Binomial Ratios and Equations
Your Name
July 22, 2024
1 Introduction
The binomial theorem provides a powerful tool for expanding expressions raised
to a power. Understanding binomial ratios involves working with binomial co-
efficients and their applications in combinatorics and probability.
2 Binomial Theorem
2.1 Theorem Statement
The binomial theorem states that for any positive integer nand any real numbers
xand y:
(x+y)n=
n
X
k=0 n
kxn−kyk
where n
kdenotes the binomial coefficient.
2.2 Binomial Coefficients
The binomial coefficient n
kis defined as:
n
k=n!
k!(n−k)!
where n! (n factorial) is the product of all positive integers up to n.
2.3 Example: Expansion
Expand (2x+ 3)3:
•Using the binomial theorem:
(2x+ 3)3=
3
X
k=0 3
k(2x)3−k
·3k
1
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
4 Applications of Binomial Coefficients
4.1 Combinatorics
Binomial coefficients are used to count combinations. For example, the number
of ways to choose kobjects from nobjects is given by n
k.
4.2 Probability
In probability, binomial coefficients are used to calculate probabilities in bino-
mial distributions. For example, the probability of getting exactly ksuccesses
in ntrials is:
P(X=k) = n
kpk(1 −p)n−k
where pis the probability of success.
5 Conclusion
The binomial theorem and binomial coefficients are fundamental concepts in
algebra and probability. Understanding binomial ratios and their applications
helps in solving various problems in combinatorics and probability.
3
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
4 Applications of Binomial Coefficients
4.1 Combinatorics
Binomial coefficients are used to count combinations. For example, the number
of ways to choose kobjects from nobjects is given by n
k.
4.2 Probability
In probability, binomial coefficients are used to calculate probabilities in bino-
mial distributions. For example, the probability of getting exactly ksuccesses
in ntrials is:
P(X=k) = n
kpk(1 −p)n−k
where pis the probability of success.
5 Conclusion
The binomial theorem and binomial coefficients are fundamental concepts in
algebra and probability. Understanding binomial ratios and their applications
helps in solving various problems in combinatorics and probability.
3
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
4 Applications of Binomial Coefficients
4.1 Combinatorics
Binomial coefficients are used to count combinations. For example, the number
of ways to choose kobjects from nobjects is given by n
k.
4.2 Probability
In probability, binomial coefficients are used to calculate probabilities in bino-
mial distributions. For example, the probability of getting exactly ksuccesses
in ntrials is:
P(X=k) = n
kpk(1 −p)n−k
where pis the probability of success.
5 Conclusion
The binomial theorem and binomial coefficients are fundamental concepts in
algebra and probability. Understanding binomial ratios and their applications
helps in solving various problems in combinatorics and probability.
3
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
•Calculate each term:
For k=0: 3
0(2x)3
·30= 1 ·8x3= 8x3,
For k=1: 3
1(2x)2
·31= 3 ·4x2
·3 = 36x2,
For k=2: 3
2(2x)1
·32= 3 ·2x·9 = 54x,
For k=3: 3
3(2x)0
·33= 1 ·27 = 27.
•Combine terms:
(2x+ 3)3= 8x3+ 36x2+ 54x+ 27
3 Binomial Ratios
3.1 Ratios of Binomial Coefficients
The ratio of binomial coefficients can be expressed as:
n
k
n
k−1=n−(k−1)
k
This ratio helps in simplifying expressions involving binomial coefficients.
3.2 Example: Ratio Calculation
Calculate the ratio of binomial coefficients for n= 5 and k= 3:
5
3
5
2
•Compute 5
3and 5
2:
5
3=5!
3!(5 −3)! =5!
3! ·2! = 10
5
2=5!
2!(5 −2)! =5!
2! ·3! = 10
•Compute the ratio:
5
3
5
2=10
10 = 1
2
4 Applications of Binomial Coefficients
4.1 Combinatorics
Binomial coefficients are used to count combinations. For example, the number
of ways to choose kobjects from nobjects is given by n
k.
4.2 Probability
In probability, binomial coefficients are used to calculate probabilities in bino-
mial distributions. For example, the probability of getting exactly ksuccesses
in ntrials is:
P(X=k) = n
kpk(1 −p)n−k
where pis the probability of success.
5 Conclusion
The binomial theorem and binomial coefficients are fundamental concepts in
algebra and probability. Understanding binomial ratios and their applications
helps in solving various problems in combinatorics and probability.
3
