MATH 201 - INTRODUCTION TO
PROBABILITY AND STATISTICS -
Combinatorial Analysis
Question Bank - Set 9
Liberty University
Question 1
Question
In a group of 10 people, how many ways are there to choose a committee of 4
people?
Solution
Step 1: To solve this problem, we will use the formula for combinations. The
number of ways to choose a committee of kpeople from a group of npeople is
given by the formula:
C(n, k) = n!
k!(n−k)!
Step 2: In this case, we want to choose a committee of 4 people from a group
of 10 people. So we have:
n= 10 and k= 4
Step 3: Plugging these values into the formula for combinations, we get:
C(10,4) = 10!
4!(10 −4)!
Step 4: Calculating the factorials in the formula, we have:
C(10,4) = 10 ×9×8×7×6×5×4×3×2×1
4×3×2×1×6×5×4×3×2×1
Step 5: Simplifying the expression, we get:
C(10,4) = 10 ×9×8×7
4×3×2×1=5040
24 = 210
Step 6: Therefore, there are 210 ways to choose a committee of 4 people
from a group of 10 people.
Question 2
Question
A committee of 5 people is to be formed from a group of 8 men and 5 women.
Find the total number of ways the committee can be formed if: a) No restrictions
are placed on who can be in the committee. b) The committee must have at
least 2 men.
Solution
a) To find the total number of ways the committee can be formed with no
restrictions on who can be in the committee, we can simply use the concept of
combinations.
• First, we find the total number of ways to choose 5 people from the group
of 13 (8 men and 5 women).
• This can be calculated as (13
5)=13!
5!(13−5)! = 1287 ways.
Step 1: Calculate the total number of ways the committee can be formed
with no restrictions:
(13
5)=13!
5!(13 −5)! = 1287 ways
b) Now, we need to find the total number of ways to form a committee with
at least 2 men.
• We can first find the total number of ways to choose 5 people without any
restrictions as we did in part (a).
• Next, we can find the number of ways to choose a committee with 0 or 1
man, and then subtract this from the total number of ways.
Step 1: Calculate the total number of ways the committee can be formed
with no restrictions:
(13
5)=13!
5!(13 −5)! = 1287 ways
Step 2: Calculate the number of ways to choose a committee with 0 or 1
man:
2
• If the committee has 0 men, we need to choose 5 women from the 5
available women, which can be done in (5
5)= 1 way.
• If the committee has 1 man, we need to choose 1 man from 8 and 4 more
people from the remaining 12, which can be done in (8
1)×(12
4)= 33×495 =
16335 ways.
• So, the total number of ways to choose a committee with 0 or 1 man is
1 + 16335 = 16336 ways.
Step 3: Calculate the number of ways to choose a committee with at least
2 men:
• The total number of ways to form a committee with at least 2 men is the
total number of ways minus the number of ways to choose a committee
with 0 or 1 man: 1287 −16336 = −15049 ways.
Therefore, the total number of ways to form a committee with at least 2
men is 15049 ways.
Question 3
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
If the committee must include at least 2 women, how many different committees
can be formed?
Solution
Step 1: Find the total number of ways to form a committee of 5 people without
any restrictions. Step 2: Find the number of ways to form a committee with
no women. Step 3: Find the number of ways to form a committee with exactly
1 woman. Step 4: Find the total number of ways to form a committee with
at least 2 women by subtracting the results from steps 2 and 3 from the total
obtained in step 1.
Step 1: The total number of ways to form a committee of 5 people from a
group of 18 is given by the combination formula:
(18
5)=18!
5!(18 −5)! =18 ×17 ×16 ×15 ×14
5×4×3×2×1= 8,568
Step 2: The number of ways to form a committee with no women is the
number of ways to choose 5 men from 10 men:
(10
5)=10!
5!(10 −5)! =10 ×9×8×7×6
5×4×3×2×1= 252
3
Step 3: The number of ways to form a committee with exactly 1 woman is
the number of ways to choose 1 woman from 8 women and 4 men from 10 men:
(8
1)×(10
4)= 8 ×210 = 1,680
Step 4: The number of ways to form a committee with at least 2 women is
given by:
8,568 −252 −1,680 = 6,636
Therefore, there are 6,636 different committees that can be formed with at
least 2 women.
Question 4
Question
In a group of 10 students, how many different ways can we select a committee of
4 students if two of the students, John and Sarah, refuse to be on the committee
together?
Solution
Step 1: First, let’s calculate the total number of ways we can select a committee
of 4 students from a group of 10 students. The total number of ways to select
a committee of 4 from 10 students is given by the combination formula:
(10
4)=10!
4!(10 −4)! =10 ×9×8×7
4×3×2×1= 210
Step 2: Next, let’s calculate the number of ways John and Sarah can be on
the committee together. Since John and Sarah refuse to be on the committee
together, we can group them and treat them as one entity. This reduces the
problem to selecting a committee of 3 from the remaining 8 students (John,
Sarah, and 8 other students). The number of ways to select a committee of 3
from 8 students is given by the combination formula:
(8
3)=8!
3!(8 −3)! =8×7×6
3×2×1= 56
Step 3: Finally, to find the number of ways to select a committee of 4
students such that John and Sarah are not together, we subtract the number of
ways they can be together from the total number of ways to select a committee:
210 −56 = 154
Therefore, there are 154 different ways to select a committee of 4 students
from a group of 10 students if John and Sarah refuse to be on the committee
together.
4
Question 5
Question
A committee of 5 people is to be formed from a group of 10 individuals, including
4 men and 6 women. How many ways can the committee be formed if it must
contain at least 2 men?
Solution
Step 1: Find the total number of ways to form a committee of 5 people. Step
2: Find the number of ways to form a committee with no men. Step 3: Find
the number of ways to form a committee with exactly 1 man. Step 4: Subtract
the results from Steps 2 and 3 from the total number of ways in Step 1 to find
the number of ways to form a committee with at least 2 men.
Step 1: Total number of ways to form a committee of 5 people from 10
individuals: (10
5)=10!
5!(10 −5)! = 252
Step 2: Number of ways to form a committee with no men: Choosing 5
women from 6: (6
5)= 6
Step 3: Number of ways to form a committee with exactly 1 man: Choose
1 man from 4 and 4 women from 6:
(4
1)×(6
4)= 4 ×15 = 60
Step 4: Number of ways to form a committee with at least 2 men:
252 −6−60 = 186
Therefore, there are 186 ways to form a committee of 5 people that must
contain at least 2 men from a group of 10 individuals.
Question 6
Question
A committee of 6 people is to be formed from a group of 10 men and 8 women.
If the committee must consist of at least 3 men and at least 2 women, how many
different committees can be formed?
5
Solution
Step 1: Calculate the number of ways to choose at least 3 men from 10 men.
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120
Step 2: Calculate the number of ways to choose at least 2 women from 8
women. (8
2)=8!
2!(8 −2)! =8×7
2×1= 28
Step 3: Calculate the total number of different committees that can be
formed. Since the committee must consist of at least 3 men and at least 2
women, we have to consider different scenarios: 1. 3 men and 2 women: 120×28
ways 2. 3 men and 3 women: (10
3)×(8
3)ways 3. 4 men and 2 women: (10
4)×(8
2)
ways 4. 4 men and 3 women: (10
4)×(8
3)ways 5. 5 men and 2 women: (10
5)×(8
2)
ways 6. 5 men and 3 women: (10
5)×(8
3)ways 7. 6 men and 2 women: (10
6)×(8
2)
ways
Therefore, the total number of different committees that can be formed is:
120×28+(10
3)×(8
3)+(10
4)×(8
2)+(10
4)×(8
3)+(10
5)×(8
2)+(10
5)×(8
3)+(10
6)×(8
2)
Calculating this expression will give us the total number of different com-
mittees that can be formed.
Question 7
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
1. In how many ways can the committee be formed if it must consist of 3
men and 2 women?
2. In how many ways can the committee be formed if it can consist of any
combination of men and women?
Solution
1. To find the number of ways the committee can be formed with 3 men and
2 women, we need to first choose the 3 men from 10 men and 2 women from
8 women. We will then multiply these two counts to get the total number of
ways.
Step 1: Calculate the number of ways to choose 3 men from 10 men.
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120
6
Step 2: Calculate the number of ways to choose 2 women from 8 women.
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28
Step 3: Multiply the results of Step 1 and Step 2 to find the total number
of ways the committee can be formed with 3 men and 2 women.
120 ×28 = 3360
Therefore, there are 3360 ways to form the committee with 3 men and 2
women.
2. To find the number of ways the committee can be formed with any
combination of men and women, we can consider all possible scenarios and add
up the different cases.
Case 1: Committee with 5 men
(10
5)= 252
Case 2: Committee with 4 men and 1 woman
(10
4)×(8
1)= 210 ×8 = 1680
Case 3: Committee with 3 men and 2 women (calculated in part 1)
3360
Case 4: Committee with 2 men and 3 women
(10
2)×(8
3)= 45 ×56 = 2520
Case 5: Committee with 1 man and 4 women
(10
1)×(8
4)= 10 ×70 = 700
Case 6: Committee with 5 women
(8
5)= 56
Step 4: Add up all the cases to find the total number of ways the committee
can be formed.
252 + 1680 + 3360 + 2520 + 700 + 56 = 7568
Therefore, there are 7568 ways to form the committee with any combination
of men and women.
7
Question 8
Question
An airline offers flights between 5 different cities. How many different one-way
routes are possible if a passenger can choose to fly directly between any two
cities without passing through any other city?
Solution
Step 1: We can treat this problem as a permutation problem since the order of
cities matters in a one-way flight.
Step 2: There are 5 cities and a passenger can choose any city as their
starting point. This means there are 5 options for the first city.
Step 3: After choosing the first city, the passenger has 4 remaining cities to
choose from for the second city (since the passenger cannot fly back to the first
city directly).
Step 4: Following the same reasoning, for the third city, there are 3 choices
left, then 2 choices for the fourth city, and finally only 1 choice for the fifth city
(since the passenger has visited all other cities).
Step 5: To find the total number of one-way routes, we multiply the number
of choices for each city in order. Therefore, the total number of possible one-way
routes is 5×4×3×2×1 = 5! = 120.
Step 6: Therefore, there are 120 different one-way routes possible when a
passenger can choose to fly directly between any two cities without passing
through any other city.
Question 9
Question
In a group of 10 people, how many ways can you choose a president, vice presi-
dent, and treasurer?
Solution
To determine the number of ways we can choose a president, vice president, and
treasurer from a group of 10 people, we can use the concept of permutations.
Step 1: Determine the number of ways to choose the president. Since there
are 10 people in the group, there are 10 choices for the president.
Step 2: Determine the number of ways to choose the vice president. After
choosing the president, there are now 9 remaining people, so there are 9 choices
for the vice president.
Step 3: Determine the number of ways to choose the treasurer. After
choosing the president and vice president, there are 8 remaining people, so
there are 8 choices for the treasurer.
8
Step 4: Calculate the total number of ways. To find the total number of
ways to choose the president, vice president, and treasurer, multiply the number
of choices for each position:
10 ×9×8 = 720
Therefore, there are 720 ways to choose a president, vice president, and
treasurer from a group of 10 people.
Question 10
Question
A company is selecting a team of 5 employees to represent them at a conference.
If the company has 12 employees to choose from, how many different ways can
the team be selected?
Solution
Step 1: Identify the number of employees to select and the total number of
employees. Let the number of employees to select be r= 5 and the total
number of employees be n= 12.
Step 2: Use the formula for combinations to find the number of ways to
select remployees from a group of nemployees. The number of ways to select
remployees from a group of nemployees is given by the formula:
C(n, r) = n!
r!(n−r)!
Step 3: Substitute n= 12 and r= 5 into the formula and calculate.
C(12,5) = 12!
5!(12 −5)!
C(12,5) = 12!
5!7!
C(12,5) = 12 ×11 ×10 ×9×8
5×4×3×2×1
C(12,5) = 792
Therefore, there are 792 different ways to select a team of 5 employees from
a group of 12 employees.
Question 11
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women. If
the committee must consist of at least 2 men and 2 women, how many different
committees can be formed?
9
Solution
Step 1: Calculate the number of ways to choose 2 men and 3 women for the
committee. There are (10
2)ways to choose 2 men from the group of 10 men, and
(8
3)ways to choose 3 women from the group of 8 women. Therefore, the number
of ways to choose 2 men and 3 women is (10
2)·(8
3).
Step 2: Calculate the number of ways to choose 3 men and 2 women for the
committee. Similarly, there are (10
3)ways to choose 3 men from the group of 10
men, and (8
2)ways to choose 2 women from the group of 8 women. Thus, the
number of ways to choose 3 men and 2 women is (10
3)·(8
2).
Step 3: Add the results from Step 1 and Step 2 to find the total number of
different committees that can be formed. The total number of committees is
the sum of the number of ways to choose 2 men and 3 women and the number
of ways to choose 3 men and 2 women:
(10
2)·(8
3)+(10
3)·(8
2)=10!
2!8! ·8!
3!5! +10!
3!7! ·8!
2!6!
= 45 ·56 + 120 ·28
= 2520 + 3360
= 5880.
Therefore, there are 5880 different committees that can be formed with at
least 2 men and 2 women.
Question 12
Question
In how many ways can you arrange the letters in the word ”LIBERTY” such
that no two vowels are adjacent?
Solution
Let’s first identify the total number of ways to arrange the letters in the word
”LIBERTY.” Then, we will calculate the number of ways that satisfy the con-
dition of no two vowels being adjacent.
Step 1: Calculate the total number of arrangements of the letters in ”LIB-
ERTY”.
The word ”LIBERTY” has 7 letters, with 3 vowels (I, E, Y) and 4 consonants
(L, B, R, T).
Using the formula for permutations of a multiset (items with repeated ele-
ments), the total number of ways to arrange the letters in ”LIBERTY” is:
7! = 5040
10
Step 2: Calculate the number of ways where no two vowels are adjacent.
To ensure that no two vowels are adjacent, we can treat the vowels (I, E, Y)
as blocks and arrange the blocks along with the consonants.
The vowels can be arranged among themselves in 3! ways.
Within each vowel block, the vowels can be rearranged in 2 ways to satisfy
the condition, with the consonants arranged in 4! ways.
So, the total number of ways where no two vowels are adjacent is:
3! ×2×4! = 6 ×24 = 144
Step 3: Calculate the final answer.
The number of ways to arrange the letters in ”LIBERTY” such that no two
vowels are adjacent is 144.
Question 13
Question
A committee of 4 students is to be formed from a group of 10 students. Two
of the students are best friends and refuse to serve on the committee together.
How many different committees can be formed?
Solution
Step 1: Find the total number of ways to form a committee without any restric-
tions. The total number of ways to select 4 students from 10 without restrictions
is given by the combination formula:
C(n, r) = n!
r!(n−r)!
Here, n= 10 (total number of students) and r= 4 (number of students in the
committee).
C(10,4) = 10!
4!(10 −4)! =10!
4!6! =10 ×9×8×7
4×3×2×1= 210
Step 2: Find the number of ways to form a committee with both best friends
included. If both best friends are included, then we have to select 2 more stu-
dents from the remaining 8 students (excluding the best friends). The number
of ways to select 2 students from 8 is given by:
C(8,2) = 8!
2!(8 −2)! =8!
2!6! =8×7
2×1= 28
Step 3: Find the number of ways to form a committee with both best friends
excluded. Since the best friends cannot serve on the committee together, we
11
need to subtract the number of committees with both best friends included from
the total number of committees.
T otal committees −Committees with both best f riends = 210 −28 = 182
Therefore, there are 182 different committees that can be formed from the
group of students.
Question 14
Question
In a group of 10 people, how many ways are there to choose a president, vice
president, and secretary?
Solution
Step 1: There are 10 ways to choose the president from the 10 people.
Step 2: After the president is chosen, there are 9 ways to choose the vice
president from the remaining 9 people.
Step 3: Finally, after the president and vice president are chosen, there are
8 ways to choose the secretary from the remaining 8 people.
So, the total number of ways to choose a president, vice president, and
secretary is:
10 ×9×8 = 720
Therefore, there are 720 ways to choose a president, vice president, and
secretary from a group of 10 people.
Question 15
Question
A committee of 5 people is to be formed from a group of 10 women and 8
men. In how many ways can the committee be formed if it must have at least
3 women?
Solution
Step 1: Find the number of ways to choose a committee with exactly 3 women.
There are (10
3)ways to choose 3 women from the 10 available, and (8
2)ways
to choose 2 men from the 8 available. So, the number of ways to choose a
committee with exactly 3 women is (10
3)·(8
2).
Step 2: Find the number of ways to choose a committee with exactly 4
women. There are (10
4)ways to choose 4 women from the 10 available, and (8
1)
ways to choose 1 man from the 8 available. So, the number of ways to choose a
committee with exactly 4 women is (10
4)·(8
1).
12
Step 3: Find the number of ways to choose a committee with exactly 5
women. There is only 1 way to choose all 5 women and no men. So, the number
of ways to choose a committee with exactly 5 women is 1.
Step 4: Add up the number of ways from Step 1, Step 2, and Step 3 to get
the total number of ways to form the committee with at least 3 women. Total
number of ways = (10
3)·(8
2)+(10
4)·(8
1)+ 1.
Calculate the values of the binomial coefficients and add them to find the
total number of ways.
Question 16
Question
In a group of 10 people, how many ways can we select a committee of 3 people
to serve as president, vice president, and treasurer?
Solution
Step 1: To find the number of ways to select the president, we have 10 choices
for the first person, 9 choices for the vice president (since we cannot select the
same person twice), and 8 choices for the treasurer. Therefore, the number of
ways to select the president, vice president, and treasurer is 10 ×9×8.
Step 2: However, the order in which we select the president, vice president,
and treasurer does not matter, so we have to divide by the number of ways to
arrange 3 people, which is 3! = 6.
Step 3: Therefore, the total number of ways to select a committee of 3 people
to serve as president, vice president, and treasurer from a group of 10 people is
given by: 10 ×9×8
3! =720
6= 120
So, there are 120 ways to select the committee.
Question 17
Question
In a committee of 5 people, how many ways can we choose a president, a vice-
president, and a treasurer from the group if no person can hold more than one
position?
Solution
Step 1: To find the number of ways to choose the president, we have 5 choices.
Step 2: After the president is chosen, there are 4 people remaining for vice-
president.
13
Step 3: Finally, after the president and vice-president are chosen, there are 3
people remaining for treasurer.
Step 4: The total number of ways to choose the positions is the product of the
number of choices for each position.
Therefore, the total number of ways to choose the president, vice-president, and
treasurer is 5×4×3 = 60.
So, there are 60 ways to choose the positions in the committee.
Question 18
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
If the committee is to consist of 2 men and 3 women, how many different com-
mittees can be formed?
Solution
Step 1: Determine the number of ways to choose 2 men from the group of 10
men. There are (10
2)ways to choose 2 men from a group of 10 men.
(10
2)=10!
2!(10 −2)! =10 ×9
2×1= 45
Step 2: Determine the number of ways to choose 3 women from the group
of 8 women. There are (8
3)ways to choose 3 women from a group of 8 women.
(8
3)=8!
3!(8 −3)! =8×7×6
3×2×1= 56
Step 3: Multiply the number of ways to choose the men and the women to
determine the total number of different committees that can be formed. Total
number of committees = Number of ways to choose men ×Number of ways to
choose women Total number of committees = 45 ×56 = 2520
Therefore, there are 2520 different committees that can be formed with 2
men and 3 women.
Question 19
Question
A committee of 6 people is to be formed from a group of 10 men and 7 women.
If the committee must consist of 3 men and 3 women, how many different ways
can this committee be formed?
14
Solution
Step 1: Find the number of ways to select 3 men from 10 men. There are (10
3)
ways to select 3 men from 10 men.
Step 2: Find the number of ways to select 3 women from 7 women. There
are (7
3)ways to select 3 women from 7 women.
Step 3: Multiply the number of ways from Step 1 and Step 2 to find the
total number of ways to form the committee. Total number of ways = (10
3)×(7
3)
Step 4: Calculate the result. Total number of ways = 10!
3!7! ×7!
3!4!
Total number of ways = 10×9×8
3×2×1×7×6×5
3×2×1
Total number of ways = 120 ×35
Total number of ways = 4200
Therefore, there are 4200 different ways to form a committee of 3 men and
3 women from a group of 10 men and 7 women.
Question 20
Question
In a class of 30 students, 15 are freshmen, 10 are sophomores, and 5 are juniors.
If 7 students are selected at random to form a committee, what is the probability
that the committee will consist of 3 freshmen, 2 sophomores, and 2 juniors?
Solution
Step 1: Find the total number of ways to select 7 students from a class of 30.
(30
7)=30!
7!(30 −7)! = 2035800
Step 2: Find the number of ways to select 3 freshmen from 15.
(15
3)=15!
3!(15 −3)! = 455
Step 3: Find the number of ways to select 2 sophomores from 10.
(10
2)=10!
2!(10 −2)! = 45
Step 4: Find the number of ways to select 2 juniors from 5.
(5
2)=5!
2!(5 −2)! = 10
Step 5: Find the total number of ways to form a committee with the specified
composition.
455 ×45 ×10 = 204750
15
Step 6: Calculate the probability by dividing the number of favorable out-
comes by the total outcomes.
P(committee with 3 freshmen, 2 sophomores, 2 juniors) = 204750
2035800 =4095
40629 ≈0.1008
Question 21
Question
A committee of 5 people is to be formed from a group of 7 men and 4 women. If
the committee is to consist of at least 2 men and 2 women, how many different
ways can the committee be formed?
Solution
Step 1: Calculate the number of ways to form a committee with exactly 2 men
and 3 women.
There are (7
2)ways to choose 2 men from 7, and (4
3)ways to choose 3 women
from 4. Multiply these two combinations to get the total number of ways to
form the committee with exactly 2 men and 3 women.
Number of ways =(7
2)×(4
3)
=7!
2!(7 −2)! ×4!
3!(4 −3)!
=7×6
2×1×4
1
= 21 ×4
= 84
Step 2: Calculate the number of ways to form a committee with at least 3
men and 2 women.
There are (7
3)ways to choose 3 men from 7, and (4
2)ways to choose 2 women
from 4. Multiply these two combinations to get the total number of ways to
form the committee with exactly 3 men and 2 women.
Number of ways =(7
3)×(4
2)
=7!
3!(7 −3)! ×4!
2!(4 −2)!
=7×6×5
3×2×1×4×3
2×1
= 35 ×6
16
= 210
Step 3: Add the results from step 1 and step 2 to get the total number of
ways to form the committee with at least 2 men and 2 women.
Total number of ways = 84 + 210
= 294
Therefore, there are 294 different ways to form the committee with at least
2 men and 2 women.
Question 22
Question
In a group of 8 people, how many ways can we form a committee of 4 people
consisting of a president, a vice president, a treasurer, and a secretary?
Solution
Step 1: To find the number of ways to choose the president, we have 8 options.
Step 2: Once the president is chosen, there are 7 remaining people to choose
from for the vice president.
Step 3: After choosing the president and vice president, there are 6 remaining
people to choose from for the treasurer.
Step 4: Finally, after choosing the president, vice president, and treasurer,
there are 5 remaining people to choose from for the secretary.
Step 5: To find the total number of ways to form the committee, we multiply
the number of choices at each step: 8×7×6×5 = 1680
Therefore, there are 1680 ways to form a committee of 4 people consisting
of a president, a vice president, a treasurer, and a secretary from a group of 8
people.
Question 23
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
Find the probability that the committee consists of 3 men and 2 women.
Solution
Step 1: Find the total number of ways to form a committee of 5 people from
a group of 18 people. Step 2: Find the number of ways to choose 3 men from
the 10 men. Step 3: Find the number of ways to choose 2 women from the 8
17
women. Step 4: Calculate the probability of choosing 3 men and 2 women for
the committee.
Step 1: The total number of ways to form a committee of 5 people from a
group of 18 people is given by the combination formula:
(18
5)=18!
5!(18 −5)! = 8,568
Step 2: The number of ways to choose 3 men from the 10 men is given by
the combination formula:
(10
3)=10!
3!(10 −3)! = 120
Step 3: The number of ways to choose 2 women from the 8 women is given
by the combination formula:
(8
2)=8!
2!(8 −2)! = 28
Step 4: The probability of choosing 3 men and 2 women for the committee
is:
Number of ways to choose 3 men and 2 women
Total number of ways to form a committee =120 ×28
8,568 =3360
8568 ≈0.393
Therefore, the probability that the committee consists of 3 men and 2 women
is approximately 0.393.
Question 24
Question
A committee of 4 people is to be formed from a group of 8 men and 6 women.
If the committee must consist of 2 men and 2 women, how many different
committees can be formed?
Solution
Step 1: Calculate the number of ways to choose 2 men from 8. Step 2: Calculate
the number of ways to choose 2 women from 6. Step 3: Multiply the results
from Step 1 and Step 2 to find the total number of different committees that
can be formed.
Step 1: There are (8
2)ways to choose 2 men from 8.
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28
18
Step 2: There are (6
2)ways to choose 2 women from 6.
(6
2)=6!
2!(6 −2)! =6×5
2×1= 15
Step 3: Multiply the results from Step 1 and Step 2. Total number of
different committees = 28 ×15 = 420
Therefore, there are 420 different committees that can be formed consisting
of 2 men and 2 women from the group.
Question 25
Question
A committee of 5 people is to be drawn from a group of 10 men and 8 women. If
the committee must consist of at least 3 men and 2 women, how many different
committees can be formed?
Solution
Step 1: Calculate the number of ways to choose 3 men out of 10.
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120.
Step 2: Calculate the number of ways to choose 2 women out of 8.
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28.
Step 3: Calculate the total number of ways to form a committee with at
least 3 men and 2 women. To find the total number of committees with at least
3 men and 2 women, we will consider 3 cases: Case 1: 3 men and 2 women:
120 ×28 ways. Case 2: 4 men and 1 woman: (10
4)×(8
1)ways. Case 3: 5 men
and 0 women: (10
5)ways.
Adding these cases together will give us the total number of committees that
can be formed.
120 ×28 + (10
4)×(8
1)+(10
5)= 3360 + 1680 + 252 = 5292.
Therefore, there are 5292 different committees that can be formed.
19
Step 5: Simplifying the expression, we get:
C(10,4) = 10 ×9×8×7
4×3×2×1=5040
24 = 210
Step 6: Therefore, there are 210 ways to choose a committee of 4 people
from a group of 10 people.
Question 2
Question
A committee of 5 people is to be formed from a group of 8 men and 5 women.
Find the total number of ways the committee can be formed if: a) No restrictions
are placed on who can be in the committee. b) The committee must have at
least 2 men.
Solution
a) To find the total number of ways the committee can be formed with no
restrictions on who can be in the committee, we can simply use the concept of
combinations.
• First, we find the total number of ways to choose 5 people from the group
of 13 (8 men and 5 women).
• This can be calculated as (13
5)=13!
5!(13−5)! = 1287 ways.
Step 1: Calculate the total number of ways the committee can be formed
with no restrictions:
(13
5)=13!
5!(13 −5)! = 1287 ways
b) Now, we need to find the total number of ways to form a committee with
at least 2 men.
• We can first find the total number of ways to choose 5 people without any
restrictions as we did in part (a).
• Next, we can find the number of ways to choose a committee with 0 or 1
man, and then subtract this from the total number of ways.
Step 1: Calculate the total number of ways the committee can be formed
with no restrictions:
(13
5)=13!
5!(13 −5)! = 1287 ways
Step 2: Calculate the number of ways to choose a committee with 0 or 1
man:
2
• If the committee has 0 men, we need to choose 5 women from the 5
available women, which can be done in (5
5)= 1 way.
• If the committee has 1 man, we need to choose 1 man from 8 and 4 more
people from the remaining 12, which can be done in (8
1)×(12
4)= 33×495 =
16335 ways.
• So, the total number of ways to choose a committee with 0 or 1 man is
1 + 16335 = 16336 ways.
Step 3: Calculate the number of ways to choose a committee with at least
2 men:
• The total number of ways to form a committee with at least 2 men is the
total number of ways minus the number of ways to choose a committee
with 0 or 1 man: 1287 −16336 = −15049 ways.
Therefore, the total number of ways to form a committee with at least 2
men is 15049 ways.
Question 3
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
If the committee must include at least 2 women, how many different committees
can be formed?
Solution
Step 1: Find the total number of ways to form a committee of 5 people without
any restrictions. Step 2: Find the number of ways to form a committee with
no women. Step 3: Find the number of ways to form a committee with exactly
1 woman. Step 4: Find the total number of ways to form a committee with
at least 2 women by subtracting the results from steps 2 and 3 from the total
obtained in step 1.
Step 1: The total number of ways to form a committee of 5 people from a
group of 18 is given by the combination formula:
(18
5)=18!
5!(18 −5)! =18 ×17 ×16 ×15 ×14
5×4×3×2×1= 8,568
Step 2: The number of ways to form a committee with no women is the
number of ways to choose 5 men from 10 men:
(10
5)=10!
5!(10 −5)! =10 ×9×8×7×6
5×4×3×2×1= 252
3
Step 3: The number of ways to form a committee with exactly 1 woman is
the number of ways to choose 1 woman from 8 women and 4 men from 10 men:
(8
1)×(10
4)= 8 ×210 = 1,680
Step 4: The number of ways to form a committee with at least 2 women is
given by:
8,568 −252 −1,680 = 6,636
Therefore, there are 6,636 different committees that can be formed with at
least 2 women.
Question 4
Question
In a group of 10 students, how many different ways can we select a committee of
4 students if two of the students, John and Sarah, refuse to be on the committee
together?
Solution
Step 1: First, let’s calculate the total number of ways we can select a committee
of 4 students from a group of 10 students. The total number of ways to select
a committee of 4 from 10 students is given by the combination formula:
(10
4)=10!
4!(10 −4)! =10 ×9×8×7
4×3×2×1= 210
Step 2: Next, let’s calculate the number of ways John and Sarah can be on
the committee together. Since John and Sarah refuse to be on the committee
together, we can group them and treat them as one entity. This reduces the
problem to selecting a committee of 3 from the remaining 8 students (John,
Sarah, and 8 other students). The number of ways to select a committee of 3
from 8 students is given by the combination formula:
(8
3)=8!
3!(8 −3)! =8×7×6
3×2×1= 56
Step 3: Finally, to find the number of ways to select a committee of 4
students such that John and Sarah are not together, we subtract the number of
ways they can be together from the total number of ways to select a committee:
210 −56 = 154
Therefore, there are 154 different ways to select a committee of 4 students
from a group of 10 students if John and Sarah refuse to be on the committee
together.
4
Question 5
Question
A committee of 5 people is to be formed from a group of 10 individuals, including
4 men and 6 women. How many ways can the committee be formed if it must
contain at least 2 men?
Solution
Step 1: Find the total number of ways to form a committee of 5 people. Step
2: Find the number of ways to form a committee with no men. Step 3: Find
the number of ways to form a committee with exactly 1 man. Step 4: Subtract
the results from Steps 2 and 3 from the total number of ways in Step 1 to find
the number of ways to form a committee with at least 2 men.
Step 1: Total number of ways to form a committee of 5 people from 10
individuals: (10
5)=10!
5!(10 −5)! = 252
Step 2: Number of ways to form a committee with no men: Choosing 5
women from 6: (6
5)= 6
Step 3: Number of ways to form a committee with exactly 1 man: Choose
1 man from 4 and 4 women from 6:
(4
1)×(6
4)= 4 ×15 = 60
Step 4: Number of ways to form a committee with at least 2 men:
252 −6−60 = 186
Therefore, there are 186 ways to form a committee of 5 people that must
contain at least 2 men from a group of 10 individuals.
Question 6
Question
A committee of 6 people is to be formed from a group of 10 men and 8 women.
If the committee must consist of at least 3 men and at least 2 women, how many
different committees can be formed?
5
Solution
Step 1: Calculate the number of ways to choose at least 3 men from 10 men.
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120
Step 2: Calculate the number of ways to choose at least 2 women from 8
women. (8
2)=8!
2!(8 −2)! =8×7
2×1= 28
Step 3: Calculate the total number of different committees that can be
formed. Since the committee must consist of at least 3 men and at least 2
women, we have to consider different scenarios: 1. 3 men and 2 women: 120×28
ways 2. 3 men and 3 women: (10
3)×(8
3)ways 3. 4 men and 2 women: (10
4)×(8
2)
ways 4. 4 men and 3 women: (10
4)×(8
3)ways 5. 5 men and 2 women: (10
5)×(8
2)
ways 6. 5 men and 3 women: (10
5)×(8
3)ways 7. 6 men and 2 women: (10
6)×(8
2)
ways
Therefore, the total number of different committees that can be formed is:
120×28+(10
3)×(8
3)+(10
4)×(8
2)+(10
4)×(8
3)+(10
5)×(8
2)+(10
5)×(8
3)+(10
6)×(8
2)
Calculating this expression will give us the total number of different com-
mittees that can be formed.
Question 7
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
1. In how many ways can the committee be formed if it must consist of 3
men and 2 women?
2. In how many ways can the committee be formed if it can consist of any
combination of men and women?
Solution
1. To find the number of ways the committee can be formed with 3 men and
2 women, we need to first choose the 3 men from 10 men and 2 women from
8 women. We will then multiply these two counts to get the total number of
ways.
Step 1: Calculate the number of ways to choose 3 men from 10 men.
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120
6
Step 2: Calculate the number of ways to choose 2 women from 8 women.
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28
Step 3: Multiply the results of Step 1 and Step 2 to find the total number
of ways the committee can be formed with 3 men and 2 women.
120 ×28 = 3360
Therefore, there are 3360 ways to form the committee with 3 men and 2
women.
2. To find the number of ways the committee can be formed with any
combination of men and women, we can consider all possible scenarios and add
up the different cases.
Case 1: Committee with 5 men
(10
5)= 252
Case 2: Committee with 4 men and 1 woman
(10
4)×(8
1)= 210 ×8 = 1680
Case 3: Committee with 3 men and 2 women (calculated in part 1)
3360
Case 4: Committee with 2 men and 3 women
(10
2)×(8
3)= 45 ×56 = 2520
Case 5: Committee with 1 man and 4 women
(10
1)×(8
4)= 10 ×70 = 700
Case 6: Committee with 5 women
(8
5)= 56
Step 4: Add up all the cases to find the total number of ways the committee
can be formed.
252 + 1680 + 3360 + 2520 + 700 + 56 = 7568
Therefore, there are 7568 ways to form the committee with any combination
of men and women.
7
Question 8
Question
An airline offers flights between 5 different cities. How many different one-way
routes are possible if a passenger can choose to fly directly between any two
cities without passing through any other city?
Solution
Step 1: We can treat this problem as a permutation problem since the order of
cities matters in a one-way flight.
Step 2: There are 5 cities and a passenger can choose any city as their
starting point. This means there are 5 options for the first city.
Step 3: After choosing the first city, the passenger has 4 remaining cities to
choose from for the second city (since the passenger cannot fly back to the first
city directly).
Step 4: Following the same reasoning, for the third city, there are 3 choices
left, then 2 choices for the fourth city, and finally only 1 choice for the fifth city
(since the passenger has visited all other cities).
Step 5: To find the total number of one-way routes, we multiply the number
of choices for each city in order. Therefore, the total number of possible one-way
routes is 5×4×3×2×1 = 5! = 120.
Step 6: Therefore, there are 120 different one-way routes possible when a
passenger can choose to fly directly between any two cities without passing
through any other city.
Question 9
Question
In a group of 10 people, how many ways can you choose a president, vice presi-
dent, and treasurer?
Solution
To determine the number of ways we can choose a president, vice president, and
treasurer from a group of 10 people, we can use the concept of permutations.
Step 1: Determine the number of ways to choose the president. Since there
are 10 people in the group, there are 10 choices for the president.
Step 2: Determine the number of ways to choose the vice president. After
choosing the president, there are now 9 remaining people, so there are 9 choices
for the vice president.
Step 3: Determine the number of ways to choose the treasurer. After
choosing the president and vice president, there are 8 remaining people, so
there are 8 choices for the treasurer.
8
Step 4: Calculate the total number of ways. To find the total number of
ways to choose the president, vice president, and treasurer, multiply the number
of choices for each position:
10 ×9×8 = 720
Therefore, there are 720 ways to choose a president, vice president, and
treasurer from a group of 10 people.
Question 10
Question
A company is selecting a team of 5 employees to represent them at a conference.
If the company has 12 employees to choose from, how many different ways can
the team be selected?
Solution
Step 1: Identify the number of employees to select and the total number of
employees. Let the number of employees to select be r= 5 and the total
number of employees be n= 12.
Step 2: Use the formula for combinations to find the number of ways to
select remployees from a group of nemployees. The number of ways to select
remployees from a group of nemployees is given by the formula:
C(n, r) = n!
r!(n−r)!
Step 3: Substitute n= 12 and r= 5 into the formula and calculate.
C(12,5) = 12!
5!(12 −5)!
C(12,5) = 12!
5!7!
C(12,5) = 12 ×11 ×10 ×9×8
5×4×3×2×1
C(12,5) = 792
Therefore, there are 792 different ways to select a team of 5 employees from
a group of 12 employees.
Question 11
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women. If
the committee must consist of at least 2 men and 2 women, how many different
committees can be formed?
9
Solution
Step 1: Calculate the number of ways to choose 2 men and 3 women for the
committee. There are (10
2)ways to choose 2 men from the group of 10 men, and
(8
3)ways to choose 3 women from the group of 8 women. Therefore, the number
of ways to choose 2 men and 3 women is (10
2)·(8
3).
Step 2: Calculate the number of ways to choose 3 men and 2 women for the
committee. Similarly, there are (10
3)ways to choose 3 men from the group of 10
men, and (8
2)ways to choose 2 women from the group of 8 women. Thus, the
number of ways to choose 3 men and 2 women is (10
3)·(8
2).
Step 3: Add the results from Step 1 and Step 2 to find the total number of
different committees that can be formed. The total number of committees is
the sum of the number of ways to choose 2 men and 3 women and the number
of ways to choose 3 men and 2 women:
(10
2)·(8
3)+(10
3)·(8
2)=10!
2!8! ·8!
3!5! +10!
3!7! ·8!
2!6!
= 45 ·56 + 120 ·28
= 2520 + 3360
= 5880.
Therefore, there are 5880 different committees that can be formed with at
least 2 men and 2 women.
Question 12
Question
In how many ways can you arrange the letters in the word ”LIBERTY” such
that no two vowels are adjacent?
Solution
Let’s first identify the total number of ways to arrange the letters in the word
”LIBERTY.” Then, we will calculate the number of ways that satisfy the con-
dition of no two vowels being adjacent.
Step 1: Calculate the total number of arrangements of the letters in ”LIB-
ERTY”.
The word ”LIBERTY” has 7 letters, with 3 vowels (I, E, Y) and 4 consonants
(L, B, R, T).
Using the formula for permutations of a multiset (items with repeated ele-
ments), the total number of ways to arrange the letters in ”LIBERTY” is:
7! = 5040
10
Step 2: Calculate the number of ways where no two vowels are adjacent.
To ensure that no two vowels are adjacent, we can treat the vowels (I, E, Y)
as blocks and arrange the blocks along with the consonants.
The vowels can be arranged among themselves in 3! ways.
Within each vowel block, the vowels can be rearranged in 2 ways to satisfy
the condition, with the consonants arranged in 4! ways.
So, the total number of ways where no two vowels are adjacent is:
3! ×2×4! = 6 ×24 = 144
Step 3: Calculate the final answer.
The number of ways to arrange the letters in ”LIBERTY” such that no two
vowels are adjacent is 144.
Question 13
Question
A committee of 4 students is to be formed from a group of 10 students. Two
of the students are best friends and refuse to serve on the committee together.
How many different committees can be formed?
Solution
Step 1: Find the total number of ways to form a committee without any restric-
tions. The total number of ways to select 4 students from 10 without restrictions
is given by the combination formula:
C(n, r) = n!
r!(n−r)!
Here, n= 10 (total number of students) and r= 4 (number of students in the
committee).
C(10,4) = 10!
4!(10 −4)! =10!
4!6! =10 ×9×8×7
4×3×2×1= 210
Step 2: Find the number of ways to form a committee with both best friends
included. If both best friends are included, then we have to select 2 more stu-
dents from the remaining 8 students (excluding the best friends). The number
of ways to select 2 students from 8 is given by:
C(8,2) = 8!
2!(8 −2)! =8!
2!6! =8×7
2×1= 28
Step 3: Find the number of ways to form a committee with both best friends
excluded. Since the best friends cannot serve on the committee together, we
11
need to subtract the number of committees with both best friends included from
the total number of committees.
T otal committees −Committees with both best f riends = 210 −28 = 182
Therefore, there are 182 different committees that can be formed from the
group of students.
Question 14
Question
In a group of 10 people, how many ways are there to choose a president, vice
president, and secretary?
Solution
Step 1: There are 10 ways to choose the president from the 10 people.
Step 2: After the president is chosen, there are 9 ways to choose the vice
president from the remaining 9 people.
Step 3: Finally, after the president and vice president are chosen, there are
8 ways to choose the secretary from the remaining 8 people.
So, the total number of ways to choose a president, vice president, and
secretary is:
10 ×9×8 = 720
Therefore, there are 720 ways to choose a president, vice president, and
secretary from a group of 10 people.
Question 15
Question
A committee of 5 people is to be formed from a group of 10 women and 8
men. In how many ways can the committee be formed if it must have at least
3 women?
Solution
Step 1: Find the number of ways to choose a committee with exactly 3 women.
There are (10
3)ways to choose 3 women from the 10 available, and (8
2)ways
to choose 2 men from the 8 available. So, the number of ways to choose a
committee with exactly 3 women is (10
3)·(8
2).
Step 2: Find the number of ways to choose a committee with exactly 4
women. There are (10
4)ways to choose 4 women from the 10 available, and (8
1)
ways to choose 1 man from the 8 available. So, the number of ways to choose a
committee with exactly 4 women is (10
4)·(8
1).
12
Step 3: Find the number of ways to choose a committee with exactly 5
women. There is only 1 way to choose all 5 women and no men. So, the number
of ways to choose a committee with exactly 5 women is 1.
Step 4: Add up the number of ways from Step 1, Step 2, and Step 3 to get
the total number of ways to form the committee with at least 3 women. Total
number of ways = (10
3)·(8
2)+(10
4)·(8
1)+ 1.
Calculate the values of the binomial coefficients and add them to find the
total number of ways.
Question 16
Question
In a group of 10 people, how many ways can we select a committee of 3 people
to serve as president, vice president, and treasurer?
Solution
Step 1: To find the number of ways to select the president, we have 10 choices
for the first person, 9 choices for the vice president (since we cannot select the
same person twice), and 8 choices for the treasurer. Therefore, the number of
ways to select the president, vice president, and treasurer is 10 ×9×8.
Step 2: However, the order in which we select the president, vice president,
and treasurer does not matter, so we have to divide by the number of ways to
arrange 3 people, which is 3! = 6.
Step 3: Therefore, the total number of ways to select a committee of 3 people
to serve as president, vice president, and treasurer from a group of 10 people is
given by: 10 ×9×8
3! =720
6= 120
So, there are 120 ways to select the committee.
Question 17
Question
In a committee of 5 people, how many ways can we choose a president, a vice-
president, and a treasurer from the group if no person can hold more than one
position?
Solution
Step 1: To find the number of ways to choose the president, we have 5 choices.
Step 2: After the president is chosen, there are 4 people remaining for vice-
president.
13
Step 3: Finally, after the president and vice-president are chosen, there are 3
people remaining for treasurer.
Step 4: The total number of ways to choose the positions is the product of the
number of choices for each position.
Therefore, the total number of ways to choose the president, vice-president, and
treasurer is 5×4×3 = 60.
So, there are 60 ways to choose the positions in the committee.
Question 18
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
If the committee is to consist of 2 men and 3 women, how many different com-
mittees can be formed?
Solution
Step 1: Determine the number of ways to choose 2 men from the group of 10
men. There are (10
2)ways to choose 2 men from a group of 10 men.
(10
2)=10!
2!(10 −2)! =10 ×9
2×1= 45
Step 2: Determine the number of ways to choose 3 women from the group
of 8 women. There are (8
3)ways to choose 3 women from a group of 8 women.
(8
3)=8!
3!(8 −3)! =8×7×6
3×2×1= 56
Step 3: Multiply the number of ways to choose the men and the women to
determine the total number of different committees that can be formed. Total
number of committees = Number of ways to choose men ×Number of ways to
choose women Total number of committees = 45 ×56 = 2520
Therefore, there are 2520 different committees that can be formed with 2
men and 3 women.
Question 19
Question
A committee of 6 people is to be formed from a group of 10 men and 7 women.
If the committee must consist of 3 men and 3 women, how many different ways
can this committee be formed?
14
Solution
Step 1: Find the number of ways to select 3 men from 10 men. There are (10
3)
ways to select 3 men from 10 men.
Step 2: Find the number of ways to select 3 women from 7 women. There
are (7
3)ways to select 3 women from 7 women.
Step 3: Multiply the number of ways from Step 1 and Step 2 to find the
total number of ways to form the committee. Total number of ways = (10
3)×(7
3)
Step 4: Calculate the result. Total number of ways = 10!
3!7! ×7!
3!4!
Total number of ways = 10×9×8
3×2×1×7×6×5
3×2×1
Total number of ways = 120 ×35
Total number of ways = 4200
Therefore, there are 4200 different ways to form a committee of 3 men and
3 women from a group of 10 men and 7 women.
Question 20
Question
In a class of 30 students, 15 are freshmen, 10 are sophomores, and 5 are juniors.
If 7 students are selected at random to form a committee, what is the probability
that the committee will consist of 3 freshmen, 2 sophomores, and 2 juniors?
Solution
Step 1: Find the total number of ways to select 7 students from a class of 30.
(30
7)=30!
7!(30 −7)! = 2035800
Step 2: Find the number of ways to select 3 freshmen from 15.
(15
3)=15!
3!(15 −3)! = 455
Step 3: Find the number of ways to select 2 sophomores from 10.
(10
2)=10!
2!(10 −2)! = 45
Step 4: Find the number of ways to select 2 juniors from 5.
(5
2)=5!
2!(5 −2)! = 10
Step 5: Find the total number of ways to form a committee with the specified
composition.
455 ×45 ×10 = 204750
15
Step 6: Calculate the probability by dividing the number of favorable out-
comes by the total outcomes.
P(committee with 3 freshmen, 2 sophomores, 2 juniors) = 204750
2035800 =4095
40629 ≈0.1008
Question 21
Question
A committee of 5 people is to be formed from a group of 7 men and 4 women. If
the committee is to consist of at least 2 men and 2 women, how many different
ways can the committee be formed?
Solution
Step 1: Calculate the number of ways to form a committee with exactly 2 men
and 3 women.
There are (7
2)ways to choose 2 men from 7, and (4
3)ways to choose 3 women
from 4. Multiply these two combinations to get the total number of ways to
form the committee with exactly 2 men and 3 women.
Number of ways =(7
2)×(4
3)
=7!
2!(7 −2)! ×4!
3!(4 −3)!
=7×6
2×1×4
1
= 21 ×4
= 84
Step 2: Calculate the number of ways to form a committee with at least 3
men and 2 women.
There are (7
3)ways to choose 3 men from 7, and (4
2)ways to choose 2 women
from 4. Multiply these two combinations to get the total number of ways to
form the committee with exactly 3 men and 2 women.
Number of ways =(7
3)×(4
2)
=7!
3!(7 −3)! ×4!
2!(4 −2)!
=7×6×5
3×2×1×4×3
2×1
= 35 ×6
16
= 210
Step 3: Add the results from step 1 and step 2 to get the total number of
ways to form the committee with at least 2 men and 2 women.
Total number of ways = 84 + 210
= 294
Therefore, there are 294 different ways to form the committee with at least
2 men and 2 women.
Question 22
Question
In a group of 8 people, how many ways can we form a committee of 4 people
consisting of a president, a vice president, a treasurer, and a secretary?
Solution
Step 1: To find the number of ways to choose the president, we have 8 options.
Step 2: Once the president is chosen, there are 7 remaining people to choose
from for the vice president.
Step 3: After choosing the president and vice president, there are 6 remaining
people to choose from for the treasurer.
Step 4: Finally, after choosing the president, vice president, and treasurer,
there are 5 remaining people to choose from for the secretary.
Step 5: To find the total number of ways to form the committee, we multiply
the number of choices at each step: 8×7×6×5 = 1680
Therefore, there are 1680 ways to form a committee of 4 people consisting
of a president, a vice president, a treasurer, and a secretary from a group of 8
people.
Question 23
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
Find the probability that the committee consists of 3 men and 2 women.
Solution
Step 1: Find the total number of ways to form a committee of 5 people from
a group of 18 people. Step 2: Find the number of ways to choose 3 men from
the 10 men. Step 3: Find the number of ways to choose 2 women from the 8
17
women. Step 4: Calculate the probability of choosing 3 men and 2 women for
the committee.
Step 1: The total number of ways to form a committee of 5 people from a
group of 18 people is given by the combination formula:
(18
5)=18!
5!(18 −5)! = 8,568
Step 2: The number of ways to choose 3 men from the 10 men is given by
the combination formula:
(10
3)=10!
3!(10 −3)! = 120
Step 3: The number of ways to choose 2 women from the 8 women is given
by the combination formula:
(8
2)=8!
2!(8 −2)! = 28
Step 4: The probability of choosing 3 men and 2 women for the committee
is:
Number of ways to choose 3 men and 2 women
Total number of ways to form a committee =120 ×28
8,568 =3360
8568 ≈0.393
Therefore, the probability that the committee consists of 3 men and 2 women
is approximately 0.393.
Question 24
Question
A committee of 4 people is to be formed from a group of 8 men and 6 women.
If the committee must consist of 2 men and 2 women, how many different
committees can be formed?
Solution
Step 1: Calculate the number of ways to choose 2 men from 8. Step 2: Calculate
the number of ways to choose 2 women from 6. Step 3: Multiply the results
from Step 1 and Step 2 to find the total number of different committees that
can be formed.
Step 1: There are (8
2)ways to choose 2 men from 8.
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28
18
Step 2: There are (6
2)ways to choose 2 women from 6.
(6
2)=6!
2!(6 −2)! =6×5
2×1= 15
Step 3: Multiply the results from Step 1 and Step 2. Total number of
different committees = 28 ×15 = 420
Therefore, there are 420 different committees that can be formed consisting
of 2 men and 2 women from the group.
Question 25
Question
A committee of 5 people is to be drawn from a group of 10 men and 8 women. If
the committee must consist of at least 3 men and 2 women, how many different
committees can be formed?
Solution
Step 1: Calculate the number of ways to choose 3 men out of 10.
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120.
Step 2: Calculate the number of ways to choose 2 women out of 8.
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28.
Step 3: Calculate the total number of ways to form a committee with at
least 3 men and 2 women. To find the total number of committees with at least
3 men and 2 women, we will consider 3 cases: Case 1: 3 men and 2 women:
120 ×28 ways. Case 2: 4 men and 1 woman: (10
4)×(8
1)ways. Case 3: 5 men
and 0 women: (10
5)ways.
Adding these cases together will give us the total number of committees that
can be formed.
120 ×28 + (10
4)×(8
1)+(10
5)= 3360 + 1680 + 252 = 5292.
Therefore, there are 5292 different committees that can be formed.
19
Step 5: Simplifying the expression, we get:
C(10,4) = 10 ×9×8×7
4×3×2×1=5040
24 = 210
Step 6: Therefore, there are 210 ways to choose a committee of 4 people
from a group of 10 people.
Question 2
Question
A committee of 5 people is to be formed from a group of 8 men and 5 women.
Find the total number of ways the committee can be formed if: a) No restrictions
are placed on who can be in the committee. b) The committee must have at
least 2 men.
Solution
a) To find the total number of ways the committee can be formed with no
restrictions on who can be in the committee, we can simply use the concept of
combinations.
• First, we find the total number of ways to choose 5 people from the group
of 13 (8 men and 5 women).
• This can be calculated as (13
5)=13!
5!(13−5)! = 1287 ways.
Step 1: Calculate the total number of ways the committee can be formed
with no restrictions:
(13
5)=13!
5!(13 −5)! = 1287 ways
b) Now, we need to find the total number of ways to form a committee with
at least 2 men.
• We can first find the total number of ways to choose 5 people without any
restrictions as we did in part (a).
• Next, we can find the number of ways to choose a committee with 0 or 1
man, and then subtract this from the total number of ways.
Step 1: Calculate the total number of ways the committee can be formed
with no restrictions:
(13
5)=13!
5!(13 −5)! = 1287 ways
Step 2: Calculate the number of ways to choose a committee with 0 or 1
man:
2
• If the committee has 0 men, we need to choose 5 women from the 5
available women, which can be done in (5
5)= 1 way.
• If the committee has 1 man, we need to choose 1 man from 8 and 4 more
people from the remaining 12, which can be done in (8
1)×(12
4)= 33×495 =
16335 ways.
• So, the total number of ways to choose a committee with 0 or 1 man is
1 + 16335 = 16336 ways.
Step 3: Calculate the number of ways to choose a committee with at least
2 men:
• The total number of ways to form a committee with at least 2 men is the
total number of ways minus the number of ways to choose a committee
with 0 or 1 man: 1287 −16336 = −15049 ways.
Therefore, the total number of ways to form a committee with at least 2
men is 15049 ways.
Question 3
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
If the committee must include at least 2 women, how many different committees
can be formed?
Solution
Step 1: Find the total number of ways to form a committee of 5 people without
any restrictions. Step 2: Find the number of ways to form a committee with
no women. Step 3: Find the number of ways to form a committee with exactly
1 woman. Step 4: Find the total number of ways to form a committee with
at least 2 women by subtracting the results from steps 2 and 3 from the total
obtained in step 1.
Step 1: The total number of ways to form a committee of 5 people from a
group of 18 is given by the combination formula:
(18
5)=18!
5!(18 −5)! =18 ×17 ×16 ×15 ×14
5×4×3×2×1= 8,568
Step 2: The number of ways to form a committee with no women is the
number of ways to choose 5 men from 10 men:
(10
5)=10!
5!(10 −5)! =10 ×9×8×7×6
5×4×3×2×1= 252
3
Step 3: The number of ways to form a committee with exactly 1 woman is
the number of ways to choose 1 woman from 8 women and 4 men from 10 men:
(8
1)×(10
4)= 8 ×210 = 1,680
Step 4: The number of ways to form a committee with at least 2 women is
given by:
8,568 −252 −1,680 = 6,636
Therefore, there are 6,636 different committees that can be formed with at
least 2 women.
Question 4
Question
In a group of 10 students, how many different ways can we select a committee of
4 students if two of the students, John and Sarah, refuse to be on the committee
together?
Solution
Step 1: First, let’s calculate the total number of ways we can select a committee
of 4 students from a group of 10 students. The total number of ways to select
a committee of 4 from 10 students is given by the combination formula:
(10
4)=10!
4!(10 −4)! =10 ×9×8×7
4×3×2×1= 210
Step 2: Next, let’s calculate the number of ways John and Sarah can be on
the committee together. Since John and Sarah refuse to be on the committee
together, we can group them and treat them as one entity. This reduces the
problem to selecting a committee of 3 from the remaining 8 students (John,
Sarah, and 8 other students). The number of ways to select a committee of 3
from 8 students is given by the combination formula:
(8
3)=8!
3!(8 −3)! =8×7×6
3×2×1= 56
Step 3: Finally, to find the number of ways to select a committee of 4
students such that John and Sarah are not together, we subtract the number of
ways they can be together from the total number of ways to select a committee:
210 −56 = 154
Therefore, there are 154 different ways to select a committee of 4 students
from a group of 10 students if John and Sarah refuse to be on the committee
together.
4
Question 5
Question
A committee of 5 people is to be formed from a group of 10 individuals, including
4 men and 6 women. How many ways can the committee be formed if it must
contain at least 2 men?
Solution
Step 1: Find the total number of ways to form a committee of 5 people. Step
2: Find the number of ways to form a committee with no men. Step 3: Find
the number of ways to form a committee with exactly 1 man. Step 4: Subtract
the results from Steps 2 and 3 from the total number of ways in Step 1 to find
the number of ways to form a committee with at least 2 men.
Step 1: Total number of ways to form a committee of 5 people from 10
individuals: (10
5)=10!
5!(10 −5)! = 252
Step 2: Number of ways to form a committee with no men: Choosing 5
women from 6: (6
5)= 6
Step 3: Number of ways to form a committee with exactly 1 man: Choose
1 man from 4 and 4 women from 6:
(4
1)×(6
4)= 4 ×15 = 60
Step 4: Number of ways to form a committee with at least 2 men:
252 −6−60 = 186
Therefore, there are 186 ways to form a committee of 5 people that must
contain at least 2 men from a group of 10 individuals.
Question 6
Question
A committee of 6 people is to be formed from a group of 10 men and 8 women.
If the committee must consist of at least 3 men and at least 2 women, how many
different committees can be formed?
5
Solution
Step 1: Calculate the number of ways to choose at least 3 men from 10 men.
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120
Step 2: Calculate the number of ways to choose at least 2 women from 8
women. (8
2)=8!
2!(8 −2)! =8×7
2×1= 28
Step 3: Calculate the total number of different committees that can be
formed. Since the committee must consist of at least 3 men and at least 2
women, we have to consider different scenarios: 1. 3 men and 2 women: 120×28
ways 2. 3 men and 3 women: (10
3)×(8
3)ways 3. 4 men and 2 women: (10
4)×(8
2)
ways 4. 4 men and 3 women: (10
4)×(8
3)ways 5. 5 men and 2 women: (10
5)×(8
2)
ways 6. 5 men and 3 women: (10
5)×(8
3)ways 7. 6 men and 2 women: (10
6)×(8
2)
ways
Therefore, the total number of different committees that can be formed is:
120×28+(10
3)×(8
3)+(10
4)×(8
2)+(10
4)×(8
3)+(10
5)×(8
2)+(10
5)×(8
3)+(10
6)×(8
2)
Calculating this expression will give us the total number of different com-
mittees that can be formed.
Question 7
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
1. In how many ways can the committee be formed if it must consist of 3
men and 2 women?
2. In how many ways can the committee be formed if it can consist of any
combination of men and women?
Solution
1. To find the number of ways the committee can be formed with 3 men and
2 women, we need to first choose the 3 men from 10 men and 2 women from
8 women. We will then multiply these two counts to get the total number of
ways.
Step 1: Calculate the number of ways to choose 3 men from 10 men.
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120
6
Step 2: Calculate the number of ways to choose 2 women from 8 women.
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28
Step 3: Multiply the results of Step 1 and Step 2 to find the total number
of ways the committee can be formed with 3 men and 2 women.
120 ×28 = 3360
Therefore, there are 3360 ways to form the committee with 3 men and 2
women.
2. To find the number of ways the committee can be formed with any
combination of men and women, we can consider all possible scenarios and add
up the different cases.
Case 1: Committee with 5 men
(10
5)= 252
Case 2: Committee with 4 men and 1 woman
(10
4)×(8
1)= 210 ×8 = 1680
Case 3: Committee with 3 men and 2 women (calculated in part 1)
3360
Case 4: Committee with 2 men and 3 women
(10
2)×(8
3)= 45 ×56 = 2520
Case 5: Committee with 1 man and 4 women
(10
1)×(8
4)= 10 ×70 = 700
Case 6: Committee with 5 women
(8
5)= 56
Step 4: Add up all the cases to find the total number of ways the committee
can be formed.
252 + 1680 + 3360 + 2520 + 700 + 56 = 7568
Therefore, there are 7568 ways to form the committee with any combination
of men and women.
7
Question 8
Question
An airline offers flights between 5 different cities. How many different one-way
routes are possible if a passenger can choose to fly directly between any two
cities without passing through any other city?
Solution
Step 1: We can treat this problem as a permutation problem since the order of
cities matters in a one-way flight.
Step 2: There are 5 cities and a passenger can choose any city as their
starting point. This means there are 5 options for the first city.
Step 3: After choosing the first city, the passenger has 4 remaining cities to
choose from for the second city (since the passenger cannot fly back to the first
city directly).
Step 4: Following the same reasoning, for the third city, there are 3 choices
left, then 2 choices for the fourth city, and finally only 1 choice for the fifth city
(since the passenger has visited all other cities).
Step 5: To find the total number of one-way routes, we multiply the number
of choices for each city in order. Therefore, the total number of possible one-way
routes is 5×4×3×2×1 = 5! = 120.
Step 6: Therefore, there are 120 different one-way routes possible when a
passenger can choose to fly directly between any two cities without passing
through any other city.
Question 9
Question
In a group of 10 people, how many ways can you choose a president, vice presi-
dent, and treasurer?
Solution
To determine the number of ways we can choose a president, vice president, and
treasurer from a group of 10 people, we can use the concept of permutations.
Step 1: Determine the number of ways to choose the president. Since there
are 10 people in the group, there are 10 choices for the president.
Step 2: Determine the number of ways to choose the vice president. After
choosing the president, there are now 9 remaining people, so there are 9 choices
for the vice president.
Step 3: Determine the number of ways to choose the treasurer. After
choosing the president and vice president, there are 8 remaining people, so
there are 8 choices for the treasurer.
8
Step 4: Calculate the total number of ways. To find the total number of
ways to choose the president, vice president, and treasurer, multiply the number
of choices for each position:
10 ×9×8 = 720
Therefore, there are 720 ways to choose a president, vice president, and
treasurer from a group of 10 people.
Question 10
Question
A company is selecting a team of 5 employees to represent them at a conference.
If the company has 12 employees to choose from, how many different ways can
the team be selected?
Solution
Step 1: Identify the number of employees to select and the total number of
employees. Let the number of employees to select be r= 5 and the total
number of employees be n= 12.
Step 2: Use the formula for combinations to find the number of ways to
select remployees from a group of nemployees. The number of ways to select
remployees from a group of nemployees is given by the formula:
C(n, r) = n!
r!(n−r)!
Step 3: Substitute n= 12 and r= 5 into the formula and calculate.
C(12,5) = 12!
5!(12 −5)!
C(12,5) = 12!
5!7!
C(12,5) = 12 ×11 ×10 ×9×8
5×4×3×2×1
C(12,5) = 792
Therefore, there are 792 different ways to select a team of 5 employees from
a group of 12 employees.
Question 11
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women. If
the committee must consist of at least 2 men and 2 women, how many different
committees can be formed?
9
Solution
Step 1: Calculate the number of ways to choose 2 men and 3 women for the
committee. There are (10
2)ways to choose 2 men from the group of 10 men, and
(8
3)ways to choose 3 women from the group of 8 women. Therefore, the number
of ways to choose 2 men and 3 women is (10
2)·(8
3).
Step 2: Calculate the number of ways to choose 3 men and 2 women for the
committee. Similarly, there are (10
3)ways to choose 3 men from the group of 10
men, and (8
2)ways to choose 2 women from the group of 8 women. Thus, the
number of ways to choose 3 men and 2 women is (10
3)·(8
2).
Step 3: Add the results from Step 1 and Step 2 to find the total number of
different committees that can be formed. The total number of committees is
the sum of the number of ways to choose 2 men and 3 women and the number
of ways to choose 3 men and 2 women:
(10
2)·(8
3)+(10
3)·(8
2)=10!
2!8! ·8!
3!5! +10!
3!7! ·8!
2!6!
= 45 ·56 + 120 ·28
= 2520 + 3360
= 5880.
Therefore, there are 5880 different committees that can be formed with at
least 2 men and 2 women.
Question 12
Question
In how many ways can you arrange the letters in the word ”LIBERTY” such
that no two vowels are adjacent?
Solution
Let’s first identify the total number of ways to arrange the letters in the word
”LIBERTY.” Then, we will calculate the number of ways that satisfy the con-
dition of no two vowels being adjacent.
Step 1: Calculate the total number of arrangements of the letters in ”LIB-
ERTY”.
The word ”LIBERTY” has 7 letters, with 3 vowels (I, E, Y) and 4 consonants
(L, B, R, T).
Using the formula for permutations of a multiset (items with repeated ele-
ments), the total number of ways to arrange the letters in ”LIBERTY” is:
7! = 5040
10
Step 2: Calculate the number of ways where no two vowels are adjacent.
To ensure that no two vowels are adjacent, we can treat the vowels (I, E, Y)
as blocks and arrange the blocks along with the consonants.
The vowels can be arranged among themselves in 3! ways.
Within each vowel block, the vowels can be rearranged in 2 ways to satisfy
the condition, with the consonants arranged in 4! ways.
So, the total number of ways where no two vowels are adjacent is:
3! ×2×4! = 6 ×24 = 144
Step 3: Calculate the final answer.
The number of ways to arrange the letters in ”LIBERTY” such that no two
vowels are adjacent is 144.
Question 13
Question
A committee of 4 students is to be formed from a group of 10 students. Two
of the students are best friends and refuse to serve on the committee together.
How many different committees can be formed?
Solution
Step 1: Find the total number of ways to form a committee without any restric-
tions. The total number of ways to select 4 students from 10 without restrictions
is given by the combination formula:
C(n, r) = n!
r!(n−r)!
Here, n= 10 (total number of students) and r= 4 (number of students in the
committee).
C(10,4) = 10!
4!(10 −4)! =10!
4!6! =10 ×9×8×7
4×3×2×1= 210
Step 2: Find the number of ways to form a committee with both best friends
included. If both best friends are included, then we have to select 2 more stu-
dents from the remaining 8 students (excluding the best friends). The number
of ways to select 2 students from 8 is given by:
C(8,2) = 8!
2!(8 −2)! =8!
2!6! =8×7
2×1= 28
Step 3: Find the number of ways to form a committee with both best friends
excluded. Since the best friends cannot serve on the committee together, we
11
need to subtract the number of committees with both best friends included from
the total number of committees.
T otal committees −Committees with both best f riends = 210 −28 = 182
Therefore, there are 182 different committees that can be formed from the
group of students.
Question 14
Question
In a group of 10 people, how many ways are there to choose a president, vice
president, and secretary?
Solution
Step 1: There are 10 ways to choose the president from the 10 people.
Step 2: After the president is chosen, there are 9 ways to choose the vice
president from the remaining 9 people.
Step 3: Finally, after the president and vice president are chosen, there are
8 ways to choose the secretary from the remaining 8 people.
So, the total number of ways to choose a president, vice president, and
secretary is:
10 ×9×8 = 720
Therefore, there are 720 ways to choose a president, vice president, and
secretary from a group of 10 people.
Question 15
Question
A committee of 5 people is to be formed from a group of 10 women and 8
men. In how many ways can the committee be formed if it must have at least
3 women?
Solution
Step 1: Find the number of ways to choose a committee with exactly 3 women.
There are (10
3)ways to choose 3 women from the 10 available, and (8
2)ways
to choose 2 men from the 8 available. So, the number of ways to choose a
committee with exactly 3 women is (10
3)·(8
2).
Step 2: Find the number of ways to choose a committee with exactly 4
women. There are (10
4)ways to choose 4 women from the 10 available, and (8
1)
ways to choose 1 man from the 8 available. So, the number of ways to choose a
committee with exactly 4 women is (10
4)·(8
1).
12
Step 3: Find the number of ways to choose a committee with exactly 5
women. There is only 1 way to choose all 5 women and no men. So, the number
of ways to choose a committee with exactly 5 women is 1.
Step 4: Add up the number of ways from Step 1, Step 2, and Step 3 to get
the total number of ways to form the committee with at least 3 women. Total
number of ways = (10
3)·(8
2)+(10
4)·(8
1)+ 1.
Calculate the values of the binomial coefficients and add them to find the
total number of ways.
Question 16
Question
In a group of 10 people, how many ways can we select a committee of 3 people
to serve as president, vice president, and treasurer?
Solution
Step 1: To find the number of ways to select the president, we have 10 choices
for the first person, 9 choices for the vice president (since we cannot select the
same person twice), and 8 choices for the treasurer. Therefore, the number of
ways to select the president, vice president, and treasurer is 10 ×9×8.
Step 2: However, the order in which we select the president, vice president,
and treasurer does not matter, so we have to divide by the number of ways to
arrange 3 people, which is 3! = 6.
Step 3: Therefore, the total number of ways to select a committee of 3 people
to serve as president, vice president, and treasurer from a group of 10 people is
given by: 10 ×9×8
3! =720
6= 120
So, there are 120 ways to select the committee.
Question 17
Question
In a committee of 5 people, how many ways can we choose a president, a vice-
president, and a treasurer from the group if no person can hold more than one
position?
Solution
Step 1: To find the number of ways to choose the president, we have 5 choices.
Step 2: After the president is chosen, there are 4 people remaining for vice-
president.
13
Step 3: Finally, after the president and vice-president are chosen, there are 3
people remaining for treasurer.
Step 4: The total number of ways to choose the positions is the product of the
number of choices for each position.
Therefore, the total number of ways to choose the president, vice-president, and
treasurer is 5×4×3 = 60.
So, there are 60 ways to choose the positions in the committee.
Question 18
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
If the committee is to consist of 2 men and 3 women, how many different com-
mittees can be formed?
Solution
Step 1: Determine the number of ways to choose 2 men from the group of 10
men. There are (10
2)ways to choose 2 men from a group of 10 men.
(10
2)=10!
2!(10 −2)! =10 ×9
2×1= 45
Step 2: Determine the number of ways to choose 3 women from the group
of 8 women. There are (8
3)ways to choose 3 women from a group of 8 women.
(8
3)=8!
3!(8 −3)! =8×7×6
3×2×1= 56
Step 3: Multiply the number of ways to choose the men and the women to
determine the total number of different committees that can be formed. Total
number of committees = Number of ways to choose men ×Number of ways to
choose women Total number of committees = 45 ×56 = 2520
Therefore, there are 2520 different committees that can be formed with 2
men and 3 women.
Question 19
Question
A committee of 6 people is to be formed from a group of 10 men and 7 women.
If the committee must consist of 3 men and 3 women, how many different ways
can this committee be formed?
14
Solution
Step 1: Find the number of ways to select 3 men from 10 men. There are (10
3)
ways to select 3 men from 10 men.
Step 2: Find the number of ways to select 3 women from 7 women. There
are (7
3)ways to select 3 women from 7 women.
Step 3: Multiply the number of ways from Step 1 and Step 2 to find the
total number of ways to form the committee. Total number of ways = (10
3)×(7
3)
Step 4: Calculate the result. Total number of ways = 10!
3!7! ×7!
3!4!
Total number of ways = 10×9×8
3×2×1×7×6×5
3×2×1
Total number of ways = 120 ×35
Total number of ways = 4200
Therefore, there are 4200 different ways to form a committee of 3 men and
3 women from a group of 10 men and 7 women.
Question 20
Question
In a class of 30 students, 15 are freshmen, 10 are sophomores, and 5 are juniors.
If 7 students are selected at random to form a committee, what is the probability
that the committee will consist of 3 freshmen, 2 sophomores, and 2 juniors?
Solution
Step 1: Find the total number of ways to select 7 students from a class of 30.
(30
7)=30!
7!(30 −7)! = 2035800
Step 2: Find the number of ways to select 3 freshmen from 15.
(15
3)=15!
3!(15 −3)! = 455
Step 3: Find the number of ways to select 2 sophomores from 10.
(10
2)=10!
2!(10 −2)! = 45
Step 4: Find the number of ways to select 2 juniors from 5.
(5
2)=5!
2!(5 −2)! = 10
Step 5: Find the total number of ways to form a committee with the specified
composition.
455 ×45 ×10 = 204750
15
Step 6: Calculate the probability by dividing the number of favorable out-
comes by the total outcomes.
P(committee with 3 freshmen, 2 sophomores, 2 juniors) = 204750
2035800 =4095
40629 ≈0.1008
Question 21
Question
A committee of 5 people is to be formed from a group of 7 men and 4 women. If
the committee is to consist of at least 2 men and 2 women, how many different
ways can the committee be formed?
Solution
Step 1: Calculate the number of ways to form a committee with exactly 2 men
and 3 women.
There are (7
2)ways to choose 2 men from 7, and (4
3)ways to choose 3 women
from 4. Multiply these two combinations to get the total number of ways to
form the committee with exactly 2 men and 3 women.
Number of ways =(7
2)×(4
3)
=7!
2!(7 −2)! ×4!
3!(4 −3)!
=7×6
2×1×4
1
= 21 ×4
= 84
Step 2: Calculate the number of ways to form a committee with at least 3
men and 2 women.
There are (7
3)ways to choose 3 men from 7, and (4
2)ways to choose 2 women
from 4. Multiply these two combinations to get the total number of ways to
form the committee with exactly 3 men and 2 women.
Number of ways =(7
3)×(4
2)
=7!
3!(7 −3)! ×4!
2!(4 −2)!
=7×6×5
3×2×1×4×3
2×1
= 35 ×6
16
= 210
Step 3: Add the results from step 1 and step 2 to get the total number of
ways to form the committee with at least 2 men and 2 women.
Total number of ways = 84 + 210
= 294
Therefore, there are 294 different ways to form the committee with at least
2 men and 2 women.
Question 22
Question
In a group of 8 people, how many ways can we form a committee of 4 people
consisting of a president, a vice president, a treasurer, and a secretary?
Solution
Step 1: To find the number of ways to choose the president, we have 8 options.
Step 2: Once the president is chosen, there are 7 remaining people to choose
from for the vice president.
Step 3: After choosing the president and vice president, there are 6 remaining
people to choose from for the treasurer.
Step 4: Finally, after choosing the president, vice president, and treasurer,
there are 5 remaining people to choose from for the secretary.
Step 5: To find the total number of ways to form the committee, we multiply
the number of choices at each step: 8×7×6×5 = 1680
Therefore, there are 1680 ways to form a committee of 4 people consisting
of a president, a vice president, a treasurer, and a secretary from a group of 8
people.
Question 23
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
Find the probability that the committee consists of 3 men and 2 women.
Solution
Step 1: Find the total number of ways to form a committee of 5 people from
a group of 18 people. Step 2: Find the number of ways to choose 3 men from
the 10 men. Step 3: Find the number of ways to choose 2 women from the 8
17
women. Step 4: Calculate the probability of choosing 3 men and 2 women for
the committee.
Step 1: The total number of ways to form a committee of 5 people from a
group of 18 people is given by the combination formula:
(18
5)=18!
5!(18 −5)! = 8,568
Step 2: The number of ways to choose 3 men from the 10 men is given by
the combination formula:
(10
3)=10!
3!(10 −3)! = 120
Step 3: The number of ways to choose 2 women from the 8 women is given
by the combination formula:
(8
2)=8!
2!(8 −2)! = 28
Step 4: The probability of choosing 3 men and 2 women for the committee
is:
Number of ways to choose 3 men and 2 women
Total number of ways to form a committee =120 ×28
8,568 =3360
8568 ≈0.393
Therefore, the probability that the committee consists of 3 men and 2 women
is approximately 0.393.
Question 24
Question
A committee of 4 people is to be formed from a group of 8 men and 6 women.
If the committee must consist of 2 men and 2 women, how many different
committees can be formed?
Solution
Step 1: Calculate the number of ways to choose 2 men from 8. Step 2: Calculate
the number of ways to choose 2 women from 6. Step 3: Multiply the results
from Step 1 and Step 2 to find the total number of different committees that
can be formed.
Step 1: There are (8
2)ways to choose 2 men from 8.
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28
18
Step 2: There are (6
2)ways to choose 2 women from 6.
(6
2)=6!
2!(6 −2)! =6×5
2×1= 15
Step 3: Multiply the results from Step 1 and Step 2. Total number of
different committees = 28 ×15 = 420
Therefore, there are 420 different committees that can be formed consisting
of 2 men and 2 women from the group.
Question 25
Question
A committee of 5 people is to be drawn from a group of 10 men and 8 women. If
the committee must consist of at least 3 men and 2 women, how many different
committees can be formed?
Solution
Step 1: Calculate the number of ways to choose 3 men out of 10.
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120.
Step 2: Calculate the number of ways to choose 2 women out of 8.
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28.
Step 3: Calculate the total number of ways to form a committee with at
least 3 men and 2 women. To find the total number of committees with at least
3 men and 2 women, we will consider 3 cases: Case 1: 3 men and 2 women:
120 ×28 ways. Case 2: 4 men and 1 woman: (10
4)×(8
1)ways. Case 3: 5 men
and 0 women: (10
5)ways.
Adding these cases together will give us the total number of committees that
can be formed.
120 ×28 + (10
4)×(8
1)+(10
5)= 3360 + 1680 + 252 = 5292.
Therefore, there are 5292 different committees that can be formed.
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