MATH 201 - INTRODUCTION TO
PROBABILITY AND STATISTICS -
Combinatorial Analysis
Question Bank - Set 5
Liberty University
Question 1
Question
How many ways are there to select 5 books from a shelf containing 10 different
books, if the order of selection doesn’t matter?
Solution
Step 1: To solve this problem, we will use the combination formula, which is
given by
(n
k)=n!
k!(n−k)!
where nis the total number of items to choose from and kis the number of
items to choose.
Step 2: In this case, we have n= 10 (total number of books on the shelf)
and k= 5 (number of books to select).
Step 3: Plugging these values into the combination formula, we get
(10
5)=10!
5!(10 −5)!
Step 4: Simplifying the expression further,
(10
5)=10 ×9×8×7×6
5×4×3×2×1
Step 5: Calculating the numerator and the denominator,
(10
5)=30240
120
Step 6: Finally, simplifying the fraction gives us
(10
5)= 252
Therefore, there are 252 ways to select 5 books from a shelf containing 10
different books when the order of selection doesn’t matter.
Question 2
Question
In a group of 12 people, how many ways can we form a committee of 4 people
where 2 of them are designated as co-chairs (i.e., order matters)?
Solution
Step 1: To find the number of ways to choose 4 people from 12 to form a
committee, we will use the combination formula (n
k)=n!
k!(n−k)! . In this case,
n= 12 and k= 4.
Number of ways to choose 4 people from 12 =(12
4)=12!
4!(12 −4)!
Step 2: Calculate the number of ways to choose 4 people from 12.
(12
4)=12!
4!(12 −4)! =12 ×11 ×10 ×9
4×3×2×1
Step 3: Simplify the expression to find the total number of ways to choose
4 people from 12.
(12
4)= 495
Step 4: For each committee of 4 people, there are 2 co-chairs to be selected.
Since order matters for the co-chairs, we can select the first co-chair in 4 ways
(since there are 4 committee members) and the second co-chair in 3 ways. The
total number of ways to select 2 co-chairs from the 4 committee members is
4×3 = 12.
Step 5: Finally, multiply the number of ways to form a committee of 4 people
with the number of ways to choose 2 co-chairs.
T otal number of ways =Number of ways to form a committee×Number of ways to choose 2 co-chairs
T otal number of ways = 495 ×12
T otal number of ways = 5940
Therefore, there are 5940 ways to form a committee of 4 people with 2
designated as co-chairs from a group of 12 people.
2
Question 3
Question
In a group of 10 people, how many ways can we select a committee of 5 people
if 2 specific people, A and B, refuse to serve on the committee together?
Solution
To find the number of ways to select a committee of 5 people from a group of 10
people where A and B refuse to serve together, we need to consider two cases:
when A is on the committee and when B is on the committee.
Case 1: A is on the committee
• Select A to be on the committee: (1
1)
• Select 3 more people from the remaining 8 (excluding B): (8
3)
There are (1
1)×(8
3)= 56 ways for A to be on the committee.
Case 2: B is on the committee
• Select B to be on the committee: (1
1)
• Select 3 more people from the remaining 8 (excluding A): (8
3)
There are (1
1)×(8
3)= 56 ways for B to be on the committee.
Therefore, the total number of ways to select a committee of 5 people with
A and B not serving together is 56 + 56 = 112.
Question 4
Question
A committee of 5 people is to be formed from a group of 8 women and 6 men.
In how many ways can the committee be formed if there must be at least 3
women on it?
Solution
Step 1: Calculate the number of ways to form a committee with exactly 3 women
and 2 men. There are (8
3)ways to choose 3 women from 8 and (6
2)ways to choose
2 men from 6. Therefore, the number of ways to form a committee with exactly
3 women and 2 men is:
(8
3)×(6
2)= 56 ×15 = 840
Step 2: Calculate the number of ways to form a committee with exactly 4
women and 1 man. There are (8
4)ways to choose 4 women from 8 and (6
1)ways
3
to choose 1 man from 6. Therefore, the number of ways to form a committee
with exactly 4 women and 1 man is:
(8
4)×(6
1)= 70 ×6 = 420
Step 3: Calculate the number of ways to form a committee with 5 women.
There are (8
5)ways to choose 5 women from 8 (since all women have to be on
the committee). Therefore, the number of ways to form a committee with 5
women is: (8
5)= 56
Step 4: Add the results from Step 1, Step 2, and Step 3 to find the total
number of ways to form the committee. Total number of ways = 840 + 420 +
56 = 1316
Therefore, the committee can be formed in 1316 different ways if there must
be at least 3 women on it.
Question 5
Question
In a group of 10 people, how many ways are there to choose a committee of
3 people if 2 specific people (Alice and Bob) refuse to be on the committee
together?
Solution
Step 1: The total number of ways to choose a committee of 3 people out of 10 is
given by the combination formula (n
r)=n!
r!(n−r)! . Thus, the total number of ways
to choose a committee of 3 people from 10 is (10
3)=10!
3!(10−3)! =10×9×8
3×2×1= 120.
Step 2: Now, we need to subtract the number of ways to choose a committee
that includes both Alice and Bob. If Alice and Bob are both on the committee,
we need to choose 1 more person from the remaining 8 people. The number of
ways to do this is (8
1)= 8.
Step 3: Finally, we need to subtract the number of ways to choose a commit-
tee that includes both Alice and Bob from the total number of ways to choose
a committee. We have 120 −8 = 112.
Therefore, there are 112 ways to choose a committee of 3 people from a
group of 10 people if Alice and Bob refuse to be on the committee together.
4
Question 6
Question
In a group of 10 people, how many ways can we choose a committee of 4 people
if two particular individuals, Alex and Beth, refuse to serve on the committee
together?
Solution
Step 1: First, we find the total number of ways to choose a committee of 4
people from the group of 10 without any restrictions. This can be calculated
using the combination formula:
(10
4)=10!
4!(10 −4)! =10 ×9×8×7
4×3×2×1= 210.
Step 2: Next, we find the number of ways to choose a committee of 4 people
where Alex and Beth are both members. This can be calculated by choosing 2
people from the remaining 8 (excluding Alex and Beth) and adding Alex and
Beth. This can be done as follows:
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28.
However, we need to take into account that Alex and Beth cannot serve on the
committee together, so we subtract this value from the total.
Step 3: Therefore, the number of ways to choose a committee of 4 people
where Alex and Beth refuse to serve together is:
210 −28 = 182.
Hence, there are 182 ways to choose a committee of 4 people from a group
of 10 people where Alex and Beth refuse to serve on the committee together.
Question 7
Question
A restaurant offers a menu consisting of 10 appetizers, 15 main courses, and
8 desserts. A customer wants to choose a meal consisting of one appetizer,
one main course, and one dessert. How many different meals can the customer
choose from?
Solution
Let’s use the Fundamental Principle of Counting to determine the total number
of different meals the customer can choose.
5
Step 1: Determine the number of choices for each category: - Number of
choices for appetizers: 10 - Number of choices for main courses: 15 - Number
of choices for desserts: 8
Step 2: Use the Fundamental Principle of Counting to find the total num-
ber of different meals: Total number of different meals = Number of choices
for appetizers ×Number of choices for main courses ×Number of choices for
desserts
Total number of different meals = 10 ×15 ×8
Total number of different meals = 1200
Therefore, the customer can choose from 1200 different meals consisting of
one appetizer, one main course, and one dessert.
Question 8
Question
In how many ways can you stack 5 red, 3 blue, and 2 green books on a shelf if
books of the same color are indistinguishable from each other?
Solution
Step 1: We can think of this problem as finding the number of ways to arrange
the colors of the books on the shelf.
Step 2: First, we arrange the different colors of the books. There are 3
different colors (red, blue, green) which means there are 3! ways to arrange the
colors.
Step 3: Next, within each color, we can arrange the books. For the red
books, there are (5
5)= 1 way to arrange them since they are indistinguishable.
Similarly, there is 1 way to arrange the blue books and 1 way to arrange the
green books.
Step 4: Combining the arrangements of the colors and the arrangements
within each color, the total number of ways to stack the books is 3! ·1·1·1 = 6
ways.
Therefore, there are 6 ways to stack the books on the shelf.
Question 9
Question
In how many ways can a committee of 5 people be chosen from a group of 10
individuals where 3 of them are professors and 7 are students?
6
Solution
Step 1: Determine the number of ways to choose 3 professors. Since there are
3 professors out of 10 individuals, we can choose 3 professors in (3
3)= 1 way.
Step 2: Determine the number of ways to choose 2 students. Since there are
7 students out of 10 individuals, we can choose 2 students in (7
2)= 21 ways.
Step 3: Multiply the number of ways from Step 1 and Step 2 to find the
total number of ways to choose a committee of 5 people. 1×21 = 21
Therefore, there are 21 ways to choose a committee of 5 people from a group
of 10 individuals where 3 are professors and 7 are students.
Question 10
Question
In a group of 10 students, how many ways can we select 4 students to form a
study group?
Solution
To solve this problem, we will use the concept of combinations (a type of com-
binatorial analysis) which helps us determine the number of ways to choose a
subset from a larger set without considering the order of selection.
Step 1: Recall that the number of ways to choose kobjects from a set of n
objects is given by the formula for combinations:
C(n, k) = n!
k!(n−k)!
Step 2: In this case, we have a group of 10 students from which we want
to select a study group of 4 students. So, we have n= 10 (total number of
students) and k= 4 (number of students to be selected).
Step 3: Now, plug in the values of nand kinto the combination formula
to find the number of ways to select a study group of 4 students from a group
of 10 students:
C(10,4) = 10!
4!(10 −4)!
Step 4: Calculate the factorials:
10! = 10 ×9×8×7×6×5×4×3×2×1
4! = 4 ×3×2×1
6! = 6 ×5×4×3×2×1
Step 5: Substitute the factorials back into the combination formula:
C(10,4) = 10 ×9×8×7×6×5×4×3×2×1
(4 ×3×2×1) ×(6 ×5×4×3×2×1)
7
Step 6: Simplify the expression:
C(10,4) = 10 ×9×8×7
4×3×2×1=5040
24 = 210
Step 7: Therefore, there are 210 ways to select 4 students from a group of
10 students to form a study group.
Question 11
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
Find the number of ways the committee can be formed if it must consist of at
least 3 men.
Solution
Step 1: Calculate the number of ways to form a committee with exactly 3 men
and 2 women. Let’s first calculate the ways to choose 3 men out of 10 and 2
women out of 8. This can be done by using the combination formula (n
r)=
n!
r!(n−r)! . So, the number of ways to choose 3 men from 10 is (10
3)=10!
3!(10−3)!
and the number of ways to choose 2 women from 8 is (8
2)=8!
2!(8−2)! .
Step 2: Calculate the total number of ways to form a committee with at
least 3 men. We can form a committee with exactly 3, 4, or 5 men. So, the
total number of ways to form a committee with at least 3 men is the sum of the
number of ways to form a committee with exactly 3, 4, or 5 men.
Step 3: Calculate the total number of ways to form a committee with at
least 3 men.
Total ways =(10
3)×(8
2)+(10
4)×(8
1)+(10
5)×(8
0)
Step 4: Perform the calculations.
(10
3)=10!
3!(10 −3)! = 120
(8
2)=8!
2!(8 −2)! = 28
(10
4)=10!
4!(10 −4)! = 210
(8
1)=8!
1!(8 −1)! = 8
(10
5)=10!
5!(10 −5)! = 252
8
(8
0)= 1
So the total number of ways to form a committee of 5 people consisting of
at least 3 men is:
120 ×28 + 210 ×8 + 252 ×1 = 3360 + 1680 + 252 = 5292
Therefore, there are 5,292 ways to form the committee with at least 3 men.
Question 12
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
In how many ways can the committee be formed if it must consist of at least 3
women?
Solution
Step 1: Calculate the number of ways to choose exactly 3 women and 2 men for
the committee. Since we need to choose exactly 3 women from 8 women and 2
men from 10 men, we can use the combination formula to calculate this.
(8
3)×(10
2)=8!
3!5! ×10!
2!8! = 56 ×45 = 2520
Step 2: Calculate the number of ways to choose exactly 4 women and 1 man
for the committee. Now, we need to choose exactly 4 women from 8 women and
1 man from 10 men.
(8
4)×(10
1)=8!
4!4! ×10!
1!9! = 70 ×10 = 700
Step 3: Calculate the number of ways to choose all 5 committee members as
women. For this case, we need to choose all 5 women from the 8 available.
(8
5)=8!
5!3! = 56
Step 4: Add up the results from Step 1, Step 2, and Step 3 to find the total
number of ways to form the committee.
2520 + 700 + 56 = 3276
Therefore, there are 3276 ways to form a committee of 5 people consisting
of at least 3 women from a group of 10 men and 8 women.
9
Question 13
Question
A committee of 5 people is to be formed from a group of 10 students (5 male
and 5 female). Find the number of ways the committee can be formed if it must
consist of at least 2 male students.
Solution
Step 1: Calculate the number of ways to choose 2, 3, 4, or 5 male students from
the group of 5 male students.
• Choose 2 male students: (5
2)= 10 ways.
• Choose 3 male students: (5
3)= 10 ways.
• Choose 4 male students: (5
4)= 5 ways.
• Choose 5 male students: (5
5)= 1 way.
Step 2: Calculate the number of ways to choose the remaining members of
the committee (from the females).
• For each combination of male students chosen in Step 1, we need to choose
the remaining members from the group of 5 female students.
Step 3: Calculate the total number of ways to form the committee.
• Add the results from Step 1 and Step 2 to find the total number of ways
to form the committee with at least 2 male students.
Therefore, the total number of ways the committee can be formed is:
(5
2)×(5
3)+(5
3)×(5
2)+(5
4)×(5
1)+(5
5)×(5
0)
Question 14
Question
A committee of 5 people is to be formed from a group of 7 men and 5 women. If
at least 2 men must be on the committee, how many different committees can
be formed?
10
Solution
Step 1: Calculate the total number of ways to form a committee without any
restrictions. There are 12 people total, and we need to choose a committee of
5. So, the total number of ways to form a committee without any restrictions
is given by the combination formula: (12
5).
Step 2: Calculate the number of ways to form a committee with no men.
Since at least 2 men are required on the committee, we must find the number
of ways to form a committee without any men. Given that there are 5 women,
the number of ways to choose 5 women from a group of 5 is (5
5).
Step 3: Calculate the number of ways to form a committee with exactly
1 man. Now, we need to find the number of ways to form a committee with
exactly 1 man. We choose 1 man from 7 men, which can be done in (7
1)ways.
The remaining 4 members of the committee must be women, chosen from a
group of 5, which can be done in (5
4)ways.
Step 4: Calculate the number of ways to form a committee with exactly 2
men. Next, we calculate the number of ways to form a committee with exactly
2 men. We choose 2 men from 7 men, which can be done in (7
2)ways. The
remaining 3 members of the committee must be women, chosen from a group
of 5, which can be done in (5
3)ways.
Step 5: Calculate the total number of ways to form a committee with at
least 2 men. To calculate the total number of ways to form a committee with
at least 2 men, we sum the number of ways to form a committee with exactly
2 men, exactly 1 man, and no men. Therefore, the total number of ways is:
(7
2)×(5
3)+(7
1)×(5
4)+(5
5).
Step 6: Perform the calculations. (7
2)= 21,(5
3)= 10,(7
1)= 7,(5
4)= 5.
Therefore, the total number of ways to form a committee with at least 2
men is: 21 ×10 + 7 ×5 + 1 = 220 + 35 + 1 = 256.
So, there are 256 different committees that can be formed.
Question 15
Question
In a group of 10 students, how many ways can we choose a committee of 4
students with a president, a vice president, and a treasurer? Assume that no
student can hold more than one position.
Solution
Step 1: To choose the president, we have 10 options.
Step 2: After choosing the president, there are 9 students left to choose from
for the vice president position.
Step 3: After choosing the president and vice president, there are 8 students
left to choose from for the treasurer position.
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Step 4: Finally, after assigning the positions of president, vice president, and
treasurer, there are 7 students left to fill the last remaining position.
Therefore, the total number of ways to choose the committee is given by:
10 ×9×8×7 = 5040
So, there are 5040 ways to choose a committee of 4 students with a president,
a vice president, and a treasurer from a group of 10 students.
Question 16
Question
In how many ways can a committee of 3 men and 4 women be formed from a
group of 10 men and 8 women?
Solution
Step 1: Calculate the number of ways to choose 3 men from 10.
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120
Step 2: Calculate the number of ways to choose 4 women from 8.
(8
4)=8!
4!(8 −4)! =8×7×6×5
4×3×2×1= 70
Step 3: Multiply the results from steps 1 and 2 to find the total number of
ways to form the committee.
120 ×70 = 8400
Therefore, there are 8400 ways to form a committee of 3 men and 4 women
from a group of 10 men and 8 women.
Question 17
Question
A committee of 5 people is to be formed from a group of 8 men and 6 women.
How many different committees can be formed if the committee must consist of
3 men and 2 women?
12
Solution
Step 1: Calculate the number of ways to choose 3 men from 8.
(8
3)=8!
3!(8 −3)! =8!
3!5! =8×7×6
3×2×1= 56
Step 2: Calculate the number of ways to choose 2 women from 6.
(6
2)=6!
2!(6 −2)! =6!
2!4! =6×5
2×1= 15
Step 3: Multiply the number of ways from Step 1 and Step 2 to get the total
number of committees that can be formed.
56 ×15 = 840
Therefore, there are 840 different committees that can be formed if the
committee must consist of 3 men and 2 women.
Question 18
Question
A selection committee needs to choose a president, a vice president, and a
secretary from a group of 10 candidates. How many different ways can these
three positions be filled if no person can hold more than one position?
Solution
Step 1: To determine the number of ways to choose the president (the first
position), we use the multiplication rule. Since there are 10 candidates and no
person can hold more than one position, there are 10 candidates eligible for the
position of president. Therefore, there are 10 ways to choose the president.
Step 2: After selecting the president, we move on to choose the vice president
(the second position). Since the president has already been chosen, there are 9
candidates remaining for the position of vice president. Therefore, there are 9
ways to choose the vice president.
Step 3: Finally, after selecting the president and vice president, there are 8
candidates left for the position of secretary. Thus, there are 8 ways to choose
the secretary.
Step 4: To find the total number of ways to fill all three positions, we multiply
the number of ways for each position: 10 ×9×8 = 720
Therefore, there are 720 different ways to fill the positions of president, vice
president, and secretary from a group of 10 candidates.
13
Question 19
Question
A committee of 5 people is to be formed from a group of 7 men and 6 women.
Find the probability that the committee contains at least 2 men.
Solution
Step 1: Find the total number of ways to form a committee of 5 people. Since
the committee can be made up of either all men, all women, or a mix of men
and women, we have to consider all possibilities. - All men: (7
5)ways to choose
5 men. - All women: (6
5)ways to choose 5 women. - Mix of men and women:
(7
4)×(6
1)ways to choose 4 men and 1 woman, or (7
3)×(6
2)ways to choose 3 men
and 2 women.
Therefore, the total number of ways to form a committee of 5 people is:
(7
5)+(6
5)+((7
4)×(6
1))+((7
3)×(6
2))
Step 2: Find the number of ways to form a committee with fewer than 2
men (0 or 1 man). - 0 men: (6
5)ways to choose 5 women. - 1 man: (7
1)×(6
4)
ways to choose 1 man and 4 women.
Therefore, the number of ways to form a committee with fewer than 2 men
is: (6
5)+((7
1)×(6
4))
Step 3: Calculate the probability of forming a committee with at least 2
men. The probability is given by:
1−Number of ways to form a committee with fewer than 2 men
Total number of ways to form a committee
Now, you can plug in the values to find the probability.
Question 20
Question
A group of 10 students are taking a multiple-choice quiz with 5 questions, each
having 4 possible answers. How many ways are there for the group of students
to respond to the quiz if each student responds to all 5 questions?
Solution
Step 1: For each question, there are 4 possible answers. Step 2: Since each of
the 10 students responds to all 5 questions, there are 45ways for each student
14
to respond to the quiz. Step 3: To find the total number of ways for the group
of 10 students to respond to the quiz, we multiply the number of ways for each
student to respond to the quiz by the number of students: 45×10. Step 4:
Calculating 45gives 1024, so the total number of ways for the group of 10
students to respond to the quiz is 1024 ×10 = 10240. Therefore, there are
10,240 ways for the group of students to respond to the quiz.
Question 21
Question
In a group of 10 people, how many ways are there to select a committee of 4
people without regard to order?
Solution
To find the number of ways to select a committee of 4 people from a group of
10 people without regards to order, we will use the combination formula.
Step 1: Determine the number of ways to choose 4 people from 10 without
regard to order. This is denoted by (10
4).
Step 2: Calculate (10
4)using the formula:
(n
k)=n!
k!(n−k)!
where nis the total number of people (10 in this case) and kis the number of
people to select (4 in this case).
Step 3: Substitute n= 10 and k= 4 into the formula:
(10
4)=10!
4!(10 −4)! =10!
4!6!
Step 4: Simplify the expression:
(10
4)=10 ×9×8×7×6!
(4 ×3×2×1) ×6! =10 ×9×8×7
4×3×2×1= 210
Therefore, there are 210 ways to select a committee of 4 people from a group
of 10 people without regard to order.
Question 22
Question
In a committee of 8 people, how many ways can we choose a president, vice-
president, and treasurer if no person can hold more than one office?
15
Solution
Step 1: Determine the number of ways to choose the president. Since there are
8 people in the committee, there are 8 choices for the president.
Step 2: After choosing the president, there are 7 people left to choose from
for the vice-president. Therefore, there are 7 choices for the vice-president.
Step 3: Finally, after choosing the president and vice-president, there are 6
people remaining for the position of treasurer. Thus, there are 6 choices for the
treasurer.
Step 4: To find the total number of ways to choose a president, vice-
president, and treasurer without anyone holding more than one office, we mul-
tiply the number of choices at each step: 8×7×6
Therefore, there are 8×7×6 = 336 ways to choose a president, vice-president,
and treasurer in a committee of 8 people where no person can hold more than
one office.
Question 23
Question
In a group of 10 students, how many ways can you choose a committee of 4
students if two particular students refuse to serve on the committee together?
Solution
Step 1: First, we find the total number of ways to choose a committee of 4
students from 10 students, which includes the two particular students who refuse
to serve together.
Step 2: This can be done using the combination formula (n
r)=n!
r!(n−r)! ,
where nis the total number of students and ris the number of students we
want to choose.
Step 3: Substituting n= 10 and r= 4 into the formula, we get: (10
4)=
10!
4!(10−4)! =10×9×8×7
4×3×2×1= 210 ways.
Step 4: Next, we find the number of ways to form a committee of 4 members
without the two particular students who refuse to serve together.
Step 5: This can be done by subtracting the number of ways to choose
a committee of 4 students from the group of 8 students (excluding the two
particular students) from the total number of ways found in Step 3.
Step 6: Using the combination formula with n= 8 and r= 4, we get:
(8
4)=8!
4!(8−4)! =8×7×6×5
4×3×2×1= 70 ways.
Step 7: Therefore, the number of ways to choose a committee of 4 students
from the group of 10 students with the condition that the two particular students
refuse to serve together is 210 −70 = 140 ways.
16
Question 24
Question
In a group of 10 people, how many ways are there to select a committee of 4
people with a president, a treasurer, and 2 regular members?
Solution
Step 1: To determine the number of ways to choose the president, we select one
person from the 10 people. There are 10 ways to choose the president.
Step 2: After selecting the president, we need to choose the treasurer. Since
the president has already been chosen, there are 9 remaining people to choose
from. There are 9 ways to choose the treasurer.
Step 3: For the 2 regular members, we need to choose 2 people from the
remaining 8 people (after selecting the president and the treasurer). We can
calculate this using combinations. The number of ways to choose 2 regular
members from 8 people is given by (8
2)=8!
2!(8−2)! = 28 ways.
Step 4: To find the total number of ways to select the committee of 4 people
with a president, a treasurer, and 2 regular members, we multiply the number
of ways for each position: 10 ×9×28 = 2520.
Therefore, there are 2520 ways to select a committee of 4 people with a
president, a treasurer, and 2 regular members from a group of 10 people.
Question 25
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 possible
committees can be formed?
Solution
Step 1: Calculate the number of ways to choose 2 men from the 10 men. There
are (10
2)ways to choose 2 men.
Step 2: Calculate the number of ways to choose 2 women from the 8 women.
There are (8
2)ways to choose 2 women.
Step 3: Calculate the number of ways to choose the remaining person (either
man or woman) for the committee. Since the committee must consist of at least
2 men and 2 women, we can either choose 3 men and 2 women or 2 men and 3
women for the committee.
For the first case: - Choose 3 men from the remaining 8 men: (8
3)ways. -
Choose 2 women from the remaining 8 women: (8
2)ways.
For the second case: - Choose 2 men from the remaining 8 men: (8
2)ways.
- Choose 3 women from the remaining 8 women: (8
3)ways.
17
Step 4: Calculate the total number of ways to form the committee. The
total number of ways to form the committee is the sum of the ways obtained
from the two cases: (10
2)×(8
2)×(8
3)+(10
2)×(8
2)×(8
3)
Simplify to find the total number of possible committees.
18
(10
5)=30240
120
Step 6: Finally, simplifying the fraction gives us
(10
5)= 252
Therefore, there are 252 ways to select 5 books from a shelf containing 10
different books when the order of selection doesn’t matter.
Question 2
Question
In a group of 12 people, how many ways can we form a committee of 4 people
where 2 of them are designated as co-chairs (i.e., order matters)?
Solution
Step 1: To find the number of ways to choose 4 people from 12 to form a
committee, we will use the combination formula (n
k)=n!
k!(n−k)! . In this case,
n= 12 and k= 4.
Number of ways to choose 4 people from 12 =(12
4)=12!
4!(12 −4)!
Step 2: Calculate the number of ways to choose 4 people from 12.
(12
4)=12!
4!(12 −4)! =12 ×11 ×10 ×9
4×3×2×1
Step 3: Simplify the expression to find the total number of ways to choose
4 people from 12.
(12
4)= 495
Step 4: For each committee of 4 people, there are 2 co-chairs to be selected.
Since order matters for the co-chairs, we can select the first co-chair in 4 ways
(since there are 4 committee members) and the second co-chair in 3 ways. The
total number of ways to select 2 co-chairs from the 4 committee members is
4×3 = 12.
Step 5: Finally, multiply the number of ways to form a committee of 4 people
with the number of ways to choose 2 co-chairs.
T otal number of ways =Number of ways to form a committee×Number of ways to choose 2 co-chairs
T otal number of ways = 495 ×12
T otal number of ways = 5940
Therefore, there are 5940 ways to form a committee of 4 people with 2
designated as co-chairs from a group of 12 people.
2
Question 3
Question
In a group of 10 people, how many ways can we select a committee of 5 people
if 2 specific people, A and B, refuse to serve on the committee together?
Solution
To find the number of ways to select a committee of 5 people from a group of 10
people where A and B refuse to serve together, we need to consider two cases:
when A is on the committee and when B is on the committee.
Case 1: A is on the committee
• Select A to be on the committee: (1
1)
• Select 3 more people from the remaining 8 (excluding B): (8
3)
There are (1
1)×(8
3)= 56 ways for A to be on the committee.
Case 2: B is on the committee
• Select B to be on the committee: (1
1)
• Select 3 more people from the remaining 8 (excluding A): (8
3)
There are (1
1)×(8
3)= 56 ways for B to be on the committee.
Therefore, the total number of ways to select a committee of 5 people with
A and B not serving together is 56 + 56 = 112.
Question 4
Question
A committee of 5 people is to be formed from a group of 8 women and 6 men.
In how many ways can the committee be formed if there must be at least 3
women on it?
Solution
Step 1: Calculate the number of ways to form a committee with exactly 3 women
and 2 men. There are (8
3)ways to choose 3 women from 8 and (6
2)ways to choose
2 men from 6. Therefore, the number of ways to form a committee with exactly
3 women and 2 men is:
(8
3)×(6
2)= 56 ×15 = 840
Step 2: Calculate the number of ways to form a committee with exactly 4
women and 1 man. There are (8
4)ways to choose 4 women from 8 and (6
1)ways
3
to choose 1 man from 6. Therefore, the number of ways to form a committee
with exactly 4 women and 1 man is:
(8
4)×(6
1)= 70 ×6 = 420
Step 3: Calculate the number of ways to form a committee with 5 women.
There are (8
5)ways to choose 5 women from 8 (since all women have to be on
the committee). Therefore, the number of ways to form a committee with 5
women is: (8
5)= 56
Step 4: Add the results from Step 1, Step 2, and Step 3 to find the total
number of ways to form the committee. Total number of ways = 840 + 420 +
56 = 1316
Therefore, the committee can be formed in 1316 different ways if there must
be at least 3 women on it.
Question 5
Question
In a group of 10 people, how many ways are there to choose a committee of
3 people if 2 specific people (Alice and Bob) refuse to be on the committee
together?
Solution
Step 1: The total number of ways to choose a committee of 3 people out of 10 is
given by the combination formula (n
r)=n!
r!(n−r)! . Thus, the total number of ways
to choose a committee of 3 people from 10 is (10
3)=10!
3!(10−3)! =10×9×8
3×2×1= 120.
Step 2: Now, we need to subtract the number of ways to choose a committee
that includes both Alice and Bob. If Alice and Bob are both on the committee,
we need to choose 1 more person from the remaining 8 people. The number of
ways to do this is (8
1)= 8.
Step 3: Finally, we need to subtract the number of ways to choose a commit-
tee that includes both Alice and Bob from the total number of ways to choose
a committee. We have 120 −8 = 112.
Therefore, there are 112 ways to choose a committee of 3 people from a
group of 10 people if Alice and Bob refuse to be on the committee together.
4
Question 6
Question
In a group of 10 people, how many ways can we choose a committee of 4 people
if two particular individuals, Alex and Beth, refuse to serve on the committee
together?
Solution
Step 1: First, we find the total number of ways to choose a committee of 4
people from the group of 10 without any restrictions. This can be calculated
using the combination formula:
(10
4)=10!
4!(10 −4)! =10 ×9×8×7
4×3×2×1= 210.
Step 2: Next, we find the number of ways to choose a committee of 4 people
where Alex and Beth are both members. This can be calculated by choosing 2
people from the remaining 8 (excluding Alex and Beth) and adding Alex and
Beth. This can be done as follows:
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28.
However, we need to take into account that Alex and Beth cannot serve on the
committee together, so we subtract this value from the total.
Step 3: Therefore, the number of ways to choose a committee of 4 people
where Alex and Beth refuse to serve together is:
210 −28 = 182.
Hence, there are 182 ways to choose a committee of 4 people from a group
of 10 people where Alex and Beth refuse to serve on the committee together.
Question 7
Question
A restaurant offers a menu consisting of 10 appetizers, 15 main courses, and
8 desserts. A customer wants to choose a meal consisting of one appetizer,
one main course, and one dessert. How many different meals can the customer
choose from?
Solution
Let’s use the Fundamental Principle of Counting to determine the total number
of different meals the customer can choose.
5
Step 1: Determine the number of choices for each category: - Number of
choices for appetizers: 10 - Number of choices for main courses: 15 - Number
of choices for desserts: 8
Step 2: Use the Fundamental Principle of Counting to find the total num-
ber of different meals: Total number of different meals = Number of choices
for appetizers ×Number of choices for main courses ×Number of choices for
desserts
Total number of different meals = 10 ×15 ×8
Total number of different meals = 1200
Therefore, the customer can choose from 1200 different meals consisting of
one appetizer, one main course, and one dessert.
Question 8
Question
In how many ways can you stack 5 red, 3 blue, and 2 green books on a shelf if
books of the same color are indistinguishable from each other?
Solution
Step 1: We can think of this problem as finding the number of ways to arrange
the colors of the books on the shelf.
Step 2: First, we arrange the different colors of the books. There are 3
different colors (red, blue, green) which means there are 3! ways to arrange the
colors.
Step 3: Next, within each color, we can arrange the books. For the red
books, there are (5
5)= 1 way to arrange them since they are indistinguishable.
Similarly, there is 1 way to arrange the blue books and 1 way to arrange the
green books.
Step 4: Combining the arrangements of the colors and the arrangements
within each color, the total number of ways to stack the books is 3! ·1·1·1 = 6
ways.
Therefore, there are 6 ways to stack the books on the shelf.
Question 9
Question
In how many ways can a committee of 5 people be chosen from a group of 10
individuals where 3 of them are professors and 7 are students?
6
Solution
Step 1: Determine the number of ways to choose 3 professors. Since there are
3 professors out of 10 individuals, we can choose 3 professors in (3
3)= 1 way.
Step 2: Determine the number of ways to choose 2 students. Since there are
7 students out of 10 individuals, we can choose 2 students in (7
2)= 21 ways.
Step 3: Multiply the number of ways from Step 1 and Step 2 to find the
total number of ways to choose a committee of 5 people. 1×21 = 21
Therefore, there are 21 ways to choose a committee of 5 people from a group
of 10 individuals where 3 are professors and 7 are students.
Question 10
Question
In a group of 10 students, how many ways can we select 4 students to form a
study group?
Solution
To solve this problem, we will use the concept of combinations (a type of com-
binatorial analysis) which helps us determine the number of ways to choose a
subset from a larger set without considering the order of selection.
Step 1: Recall that the number of ways to choose kobjects from a set of n
objects is given by the formula for combinations:
C(n, k) = n!
k!(n−k)!
Step 2: In this case, we have a group of 10 students from which we want
to select a study group of 4 students. So, we have n= 10 (total number of
students) and k= 4 (number of students to be selected).
Step 3: Now, plug in the values of nand kinto the combination formula
to find the number of ways to select a study group of 4 students from a group
of 10 students:
C(10,4) = 10!
4!(10 −4)!
Step 4: Calculate the factorials:
10! = 10 ×9×8×7×6×5×4×3×2×1
4! = 4 ×3×2×1
6! = 6 ×5×4×3×2×1
Step 5: Substitute the factorials back into the combination formula:
C(10,4) = 10 ×9×8×7×6×5×4×3×2×1
(4 ×3×2×1) ×(6 ×5×4×3×2×1)
7
Step 6: Simplify the expression:
C(10,4) = 10 ×9×8×7
4×3×2×1=5040
24 = 210
Step 7: Therefore, there are 210 ways to select 4 students from a group of
10 students to form a study group.
Question 11
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
Find the number of ways the committee can be formed if it must consist of at
least 3 men.
Solution
Step 1: Calculate the number of ways to form a committee with exactly 3 men
and 2 women. Let’s first calculate the ways to choose 3 men out of 10 and 2
women out of 8. This can be done by using the combination formula (n
r)=
n!
r!(n−r)! . So, the number of ways to choose 3 men from 10 is (10
3)=10!
3!(10−3)!
and the number of ways to choose 2 women from 8 is (8
2)=8!
2!(8−2)! .
Step 2: Calculate the total number of ways to form a committee with at
least 3 men. We can form a committee with exactly 3, 4, or 5 men. So, the
total number of ways to form a committee with at least 3 men is the sum of the
number of ways to form a committee with exactly 3, 4, or 5 men.
Step 3: Calculate the total number of ways to form a committee with at
least 3 men.
Total ways =(10
3)×(8
2)+(10
4)×(8
1)+(10
5)×(8
0)
Step 4: Perform the calculations.
(10
3)=10!
3!(10 −3)! = 120
(8
2)=8!
2!(8 −2)! = 28
(10
4)=10!
4!(10 −4)! = 210
(8
1)=8!
1!(8 −1)! = 8
(10
5)=10!
5!(10 −5)! = 252
8
(8
0)= 1
So the total number of ways to form a committee of 5 people consisting of
at least 3 men is:
120 ×28 + 210 ×8 + 252 ×1 = 3360 + 1680 + 252 = 5292
Therefore, there are 5,292 ways to form the committee with at least 3 men.
Question 12
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
In how many ways can the committee be formed if it must consist of at least 3
women?
Solution
Step 1: Calculate the number of ways to choose exactly 3 women and 2 men for
the committee. Since we need to choose exactly 3 women from 8 women and 2
men from 10 men, we can use the combination formula to calculate this.
(8
3)×(10
2)=8!
3!5! ×10!
2!8! = 56 ×45 = 2520
Step 2: Calculate the number of ways to choose exactly 4 women and 1 man
for the committee. Now, we need to choose exactly 4 women from 8 women and
1 man from 10 men.
(8
4)×(10
1)=8!
4!4! ×10!
1!9! = 70 ×10 = 700
Step 3: Calculate the number of ways to choose all 5 committee members as
women. For this case, we need to choose all 5 women from the 8 available.
(8
5)=8!
5!3! = 56
Step 4: Add up the results from Step 1, Step 2, and Step 3 to find the total
number of ways to form the committee.
2520 + 700 + 56 = 3276
Therefore, there are 3276 ways to form a committee of 5 people consisting
of at least 3 women from a group of 10 men and 8 women.
9
Question 13
Question
A committee of 5 people is to be formed from a group of 10 students (5 male
and 5 female). Find the number of ways the committee can be formed if it must
consist of at least 2 male students.
Solution
Step 1: Calculate the number of ways to choose 2, 3, 4, or 5 male students from
the group of 5 male students.
• Choose 2 male students: (5
2)= 10 ways.
• Choose 3 male students: (5
3)= 10 ways.
• Choose 4 male students: (5
4)= 5 ways.
• Choose 5 male students: (5
5)= 1 way.
Step 2: Calculate the number of ways to choose the remaining members of
the committee (from the females).
• For each combination of male students chosen in Step 1, we need to choose
the remaining members from the group of 5 female students.
Step 3: Calculate the total number of ways to form the committee.
• Add the results from Step 1 and Step 2 to find the total number of ways
to form the committee with at least 2 male students.
Therefore, the total number of ways the committee can be formed is:
(5
2)×(5
3)+(5
3)×(5
2)+(5
4)×(5
1)+(5
5)×(5
0)
Question 14
Question
A committee of 5 people is to be formed from a group of 7 men and 5 women. If
at least 2 men must be on the committee, how many different committees can
be formed?
10
Solution
Step 1: Calculate the total number of ways to form a committee without any
restrictions. There are 12 people total, and we need to choose a committee of
5. So, the total number of ways to form a committee without any restrictions
is given by the combination formula: (12
5).
Step 2: Calculate the number of ways to form a committee with no men.
Since at least 2 men are required on the committee, we must find the number
of ways to form a committee without any men. Given that there are 5 women,
the number of ways to choose 5 women from a group of 5 is (5
5).
Step 3: Calculate the number of ways to form a committee with exactly
1 man. Now, we need to find the number of ways to form a committee with
exactly 1 man. We choose 1 man from 7 men, which can be done in (7
1)ways.
The remaining 4 members of the committee must be women, chosen from a
group of 5, which can be done in (5
4)ways.
Step 4: Calculate the number of ways to form a committee with exactly 2
men. Next, we calculate the number of ways to form a committee with exactly
2 men. We choose 2 men from 7 men, which can be done in (7
2)ways. The
remaining 3 members of the committee must be women, chosen from a group
of 5, which can be done in (5
3)ways.
Step 5: Calculate the total number of ways to form a committee with at
least 2 men. To calculate the total number of ways to form a committee with
at least 2 men, we sum the number of ways to form a committee with exactly
2 men, exactly 1 man, and no men. Therefore, the total number of ways is:
(7
2)×(5
3)+(7
1)×(5
4)+(5
5).
Step 6: Perform the calculations. (7
2)= 21,(5
3)= 10,(7
1)= 7,(5
4)= 5.
Therefore, the total number of ways to form a committee with at least 2
men is: 21 ×10 + 7 ×5 + 1 = 220 + 35 + 1 = 256.
So, there are 256 different committees that can be formed.
Question 15
Question
In a group of 10 students, how many ways can we choose a committee of 4
students with a president, a vice president, and a treasurer? Assume that no
student can hold more than one position.
Solution
Step 1: To choose the president, we have 10 options.
Step 2: After choosing the president, there are 9 students left to choose from
for the vice president position.
Step 3: After choosing the president and vice president, there are 8 students
left to choose from for the treasurer position.
11
Step 4: Finally, after assigning the positions of president, vice president, and
treasurer, there are 7 students left to fill the last remaining position.
Therefore, the total number of ways to choose the committee is given by:
10 ×9×8×7 = 5040
So, there are 5040 ways to choose a committee of 4 students with a president,
a vice president, and a treasurer from a group of 10 students.
Question 16
Question
In how many ways can a committee of 3 men and 4 women be formed from a
group of 10 men and 8 women?
Solution
Step 1: Calculate the number of ways to choose 3 men from 10.
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120
Step 2: Calculate the number of ways to choose 4 women from 8.
(8
4)=8!
4!(8 −4)! =8×7×6×5
4×3×2×1= 70
Step 3: Multiply the results from steps 1 and 2 to find the total number of
ways to form the committee.
120 ×70 = 8400
Therefore, there are 8400 ways to form a committee of 3 men and 4 women
from a group of 10 men and 8 women.
Question 17
Question
A committee of 5 people is to be formed from a group of 8 men and 6 women.
How many different committees can be formed if the committee must consist of
3 men and 2 women?
12
Solution
Step 1: Calculate the number of ways to choose 3 men from 8.
(8
3)=8!
3!(8 −3)! =8!
3!5! =8×7×6
3×2×1= 56
Step 2: Calculate the number of ways to choose 2 women from 6.
(6
2)=6!
2!(6 −2)! =6!
2!4! =6×5
2×1= 15
Step 3: Multiply the number of ways from Step 1 and Step 2 to get the total
number of committees that can be formed.
56 ×15 = 840
Therefore, there are 840 different committees that can be formed if the
committee must consist of 3 men and 2 women.
Question 18
Question
A selection committee needs to choose a president, a vice president, and a
secretary from a group of 10 candidates. How many different ways can these
three positions be filled if no person can hold more than one position?
Solution
Step 1: To determine the number of ways to choose the president (the first
position), we use the multiplication rule. Since there are 10 candidates and no
person can hold more than one position, there are 10 candidates eligible for the
position of president. Therefore, there are 10 ways to choose the president.
Step 2: After selecting the president, we move on to choose the vice president
(the second position). Since the president has already been chosen, there are 9
candidates remaining for the position of vice president. Therefore, there are 9
ways to choose the vice president.
Step 3: Finally, after selecting the president and vice president, there are 8
candidates left for the position of secretary. Thus, there are 8 ways to choose
the secretary.
Step 4: To find the total number of ways to fill all three positions, we multiply
the number of ways for each position: 10 ×9×8 = 720
Therefore, there are 720 different ways to fill the positions of president, vice
president, and secretary from a group of 10 candidates.
13
Question 19
Question
A committee of 5 people is to be formed from a group of 7 men and 6 women.
Find the probability that the committee contains at least 2 men.
Solution
Step 1: Find the total number of ways to form a committee of 5 people. Since
the committee can be made up of either all men, all women, or a mix of men
and women, we have to consider all possibilities. - All men: (7
5)ways to choose
5 men. - All women: (6
5)ways to choose 5 women. - Mix of men and women:
(7
4)×(6
1)ways to choose 4 men and 1 woman, or (7
3)×(6
2)ways to choose 3 men
and 2 women.
Therefore, the total number of ways to form a committee of 5 people is:
(7
5)+(6
5)+((7
4)×(6
1))+((7
3)×(6
2))
Step 2: Find the number of ways to form a committee with fewer than 2
men (0 or 1 man). - 0 men: (6
5)ways to choose 5 women. - 1 man: (7
1)×(6
4)
ways to choose 1 man and 4 women.
Therefore, the number of ways to form a committee with fewer than 2 men
is: (6
5)+((7
1)×(6
4))
Step 3: Calculate the probability of forming a committee with at least 2
men. The probability is given by:
1−Number of ways to form a committee with fewer than 2 men
Total number of ways to form a committee
Now, you can plug in the values to find the probability.
Question 20
Question
A group of 10 students are taking a multiple-choice quiz with 5 questions, each
having 4 possible answers. How many ways are there for the group of students
to respond to the quiz if each student responds to all 5 questions?
Solution
Step 1: For each question, there are 4 possible answers. Step 2: Since each of
the 10 students responds to all 5 questions, there are 45ways for each student
14
to respond to the quiz. Step 3: To find the total number of ways for the group
of 10 students to respond to the quiz, we multiply the number of ways for each
student to respond to the quiz by the number of students: 45×10. Step 4:
Calculating 45gives 1024, so the total number of ways for the group of 10
students to respond to the quiz is 1024 ×10 = 10240. Therefore, there are
10,240 ways for the group of students to respond to the quiz.
Question 21
Question
In a group of 10 people, how many ways are there to select a committee of 4
people without regard to order?
Solution
To find the number of ways to select a committee of 4 people from a group of
10 people without regards to order, we will use the combination formula.
Step 1: Determine the number of ways to choose 4 people from 10 without
regard to order. This is denoted by (10
4).
Step 2: Calculate (10
4)using the formula:
(n
k)=n!
k!(n−k)!
where nis the total number of people (10 in this case) and kis the number of
people to select (4 in this case).
Step 3: Substitute n= 10 and k= 4 into the formula:
(10
4)=10!
4!(10 −4)! =10!
4!6!
Step 4: Simplify the expression:
(10
4)=10 ×9×8×7×6!
(4 ×3×2×1) ×6! =10 ×9×8×7
4×3×2×1= 210
Therefore, there are 210 ways to select a committee of 4 people from a group
of 10 people without regard to order.
Question 22
Question
In a committee of 8 people, how many ways can we choose a president, vice-
president, and treasurer if no person can hold more than one office?
15
Solution
Step 1: Determine the number of ways to choose the president. Since there are
8 people in the committee, there are 8 choices for the president.
Step 2: After choosing the president, there are 7 people left to choose from
for the vice-president. Therefore, there are 7 choices for the vice-president.
Step 3: Finally, after choosing the president and vice-president, there are 6
people remaining for the position of treasurer. Thus, there are 6 choices for the
treasurer.
Step 4: To find the total number of ways to choose a president, vice-
president, and treasurer without anyone holding more than one office, we mul-
tiply the number of choices at each step: 8×7×6
Therefore, there are 8×7×6 = 336 ways to choose a president, vice-president,
and treasurer in a committee of 8 people where no person can hold more than
one office.
Question 23
Question
In a group of 10 students, how many ways can you choose a committee of 4
students if two particular students refuse to serve on the committee together?
Solution
Step 1: First, we find the total number of ways to choose a committee of 4
students from 10 students, which includes the two particular students who refuse
to serve together.
Step 2: This can be done using the combination formula (n
r)=n!
r!(n−r)! ,
where nis the total number of students and ris the number of students we
want to choose.
Step 3: Substituting n= 10 and r= 4 into the formula, we get: (10
4)=
10!
4!(10−4)! =10×9×8×7
4×3×2×1= 210 ways.
Step 4: Next, we find the number of ways to form a committee of 4 members
without the two particular students who refuse to serve together.
Step 5: This can be done by subtracting the number of ways to choose
a committee of 4 students from the group of 8 students (excluding the two
particular students) from the total number of ways found in Step 3.
Step 6: Using the combination formula with n= 8 and r= 4, we get:
(8
4)=8!
4!(8−4)! =8×7×6×5
4×3×2×1= 70 ways.
Step 7: Therefore, the number of ways to choose a committee of 4 students
from the group of 10 students with the condition that the two particular students
refuse to serve together is 210 −70 = 140 ways.
16
Question 24
Question
In a group of 10 people, how many ways are there to select a committee of 4
people with a president, a treasurer, and 2 regular members?
Solution
Step 1: To determine the number of ways to choose the president, we select one
person from the 10 people. There are 10 ways to choose the president.
Step 2: After selecting the president, we need to choose the treasurer. Since
the president has already been chosen, there are 9 remaining people to choose
from. There are 9 ways to choose the treasurer.
Step 3: For the 2 regular members, we need to choose 2 people from the
remaining 8 people (after selecting the president and the treasurer). We can
calculate this using combinations. The number of ways to choose 2 regular
members from 8 people is given by (8
2)=8!
2!(8−2)! = 28 ways.
Step 4: To find the total number of ways to select the committee of 4 people
with a president, a treasurer, and 2 regular members, we multiply the number
of ways for each position: 10 ×9×28 = 2520.
Therefore, there are 2520 ways to select a committee of 4 people with a
president, a treasurer, and 2 regular members from a group of 10 people.
Question 25
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 possible
committees can be formed?
Solution
Step 1: Calculate the number of ways to choose 2 men from the 10 men. There
are (10
2)ways to choose 2 men.
Step 2: Calculate the number of ways to choose 2 women from the 8 women.
There are (8
2)ways to choose 2 women.
Step 3: Calculate the number of ways to choose the remaining person (either
man or woman) for the committee. Since the committee must consist of at least
2 men and 2 women, we can either choose 3 men and 2 women or 2 men and 3
women for the committee.
For the first case: - Choose 3 men from the remaining 8 men: (8
3)ways. -
Choose 2 women from the remaining 8 women: (8
2)ways.
For the second case: - Choose 2 men from the remaining 8 men: (8
2)ways.
- Choose 3 women from the remaining 8 women: (8
3)ways.
17
Step 4: Calculate the total number of ways to form the committee. The
total number of ways to form the committee is the sum of the ways obtained
from the two cases: (10
2)×(8
2)×(8
3)+(10
2)×(8
2)×(8
3)
Simplify to find the total number of possible committees.
18
(10
5)=30240
120
Step 6: Finally, simplifying the fraction gives us
(10
5)= 252
Therefore, there are 252 ways to select 5 books from a shelf containing 10
different books when the order of selection doesn’t matter.
Question 2
Question
In a group of 12 people, how many ways can we form a committee of 4 people
where 2 of them are designated as co-chairs (i.e., order matters)?
Solution
Step 1: To find the number of ways to choose 4 people from 12 to form a
committee, we will use the combination formula (n
k)=n!
k!(n−k)! . In this case,
n= 12 and k= 4.
Number of ways to choose 4 people from 12 =(12
4)=12!
4!(12 −4)!
Step 2: Calculate the number of ways to choose 4 people from 12.
(12
4)=12!
4!(12 −4)! =12 ×11 ×10 ×9
4×3×2×1
Step 3: Simplify the expression to find the total number of ways to choose
4 people from 12.
(12
4)= 495
Step 4: For each committee of 4 people, there are 2 co-chairs to be selected.
Since order matters for the co-chairs, we can select the first co-chair in 4 ways
(since there are 4 committee members) and the second co-chair in 3 ways. The
total number of ways to select 2 co-chairs from the 4 committee members is
4×3 = 12.
Step 5: Finally, multiply the number of ways to form a committee of 4 people
with the number of ways to choose 2 co-chairs.
T otal number of ways =Number of ways to form a committee×Number of ways to choose 2 co-chairs
T otal number of ways = 495 ×12
T otal number of ways = 5940
Therefore, there are 5940 ways to form a committee of 4 people with 2
designated as co-chairs from a group of 12 people.
2
Question 3
Question
In a group of 10 people, how many ways can we select a committee of 5 people
if 2 specific people, A and B, refuse to serve on the committee together?
Solution
To find the number of ways to select a committee of 5 people from a group of 10
people where A and B refuse to serve together, we need to consider two cases:
when A is on the committee and when B is on the committee.
Case 1: A is on the committee
• Select A to be on the committee: (1
1)
• Select 3 more people from the remaining 8 (excluding B): (8
3)
There are (1
1)×(8
3)= 56 ways for A to be on the committee.
Case 2: B is on the committee
• Select B to be on the committee: (1
1)
• Select 3 more people from the remaining 8 (excluding A): (8
3)
There are (1
1)×(8
3)= 56 ways for B to be on the committee.
Therefore, the total number of ways to select a committee of 5 people with
A and B not serving together is 56 + 56 = 112.
Question 4
Question
A committee of 5 people is to be formed from a group of 8 women and 6 men.
In how many ways can the committee be formed if there must be at least 3
women on it?
Solution
Step 1: Calculate the number of ways to form a committee with exactly 3 women
and 2 men. There are (8
3)ways to choose 3 women from 8 and (6
2)ways to choose
2 men from 6. Therefore, the number of ways to form a committee with exactly
3 women and 2 men is:
(8
3)×(6
2)= 56 ×15 = 840
Step 2: Calculate the number of ways to form a committee with exactly 4
women and 1 man. There are (8
4)ways to choose 4 women from 8 and (6
1)ways
3
to choose 1 man from 6. Therefore, the number of ways to form a committee
with exactly 4 women and 1 man is:
(8
4)×(6
1)= 70 ×6 = 420
Step 3: Calculate the number of ways to form a committee with 5 women.
There are (8
5)ways to choose 5 women from 8 (since all women have to be on
the committee). Therefore, the number of ways to form a committee with 5
women is: (8
5)= 56
Step 4: Add the results from Step 1, Step 2, and Step 3 to find the total
number of ways to form the committee. Total number of ways = 840 + 420 +
56 = 1316
Therefore, the committee can be formed in 1316 different ways if there must
be at least 3 women on it.
Question 5
Question
In a group of 10 people, how many ways are there to choose a committee of
3 people if 2 specific people (Alice and Bob) refuse to be on the committee
together?
Solution
Step 1: The total number of ways to choose a committee of 3 people out of 10 is
given by the combination formula (n
r)=n!
r!(n−r)! . Thus, the total number of ways
to choose a committee of 3 people from 10 is (10
3)=10!
3!(10−3)! =10×9×8
3×2×1= 120.
Step 2: Now, we need to subtract the number of ways to choose a committee
that includes both Alice and Bob. If Alice and Bob are both on the committee,
we need to choose 1 more person from the remaining 8 people. The number of
ways to do this is (8
1)= 8.
Step 3: Finally, we need to subtract the number of ways to choose a commit-
tee that includes both Alice and Bob from the total number of ways to choose
a committee. We have 120 −8 = 112.
Therefore, there are 112 ways to choose a committee of 3 people from a
group of 10 people if Alice and Bob refuse to be on the committee together.
4
Question 6
Question
In a group of 10 people, how many ways can we choose a committee of 4 people
if two particular individuals, Alex and Beth, refuse to serve on the committee
together?
Solution
Step 1: First, we find the total number of ways to choose a committee of 4
people from the group of 10 without any restrictions. This can be calculated
using the combination formula:
(10
4)=10!
4!(10 −4)! =10 ×9×8×7
4×3×2×1= 210.
Step 2: Next, we find the number of ways to choose a committee of 4 people
where Alex and Beth are both members. This can be calculated by choosing 2
people from the remaining 8 (excluding Alex and Beth) and adding Alex and
Beth. This can be done as follows:
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28.
However, we need to take into account that Alex and Beth cannot serve on the
committee together, so we subtract this value from the total.
Step 3: Therefore, the number of ways to choose a committee of 4 people
where Alex and Beth refuse to serve together is:
210 −28 = 182.
Hence, there are 182 ways to choose a committee of 4 people from a group
of 10 people where Alex and Beth refuse to serve on the committee together.
Question 7
Question
A restaurant offers a menu consisting of 10 appetizers, 15 main courses, and
8 desserts. A customer wants to choose a meal consisting of one appetizer,
one main course, and one dessert. How many different meals can the customer
choose from?
Solution
Let’s use the Fundamental Principle of Counting to determine the total number
of different meals the customer can choose.
5
Step 1: Determine the number of choices for each category: - Number of
choices for appetizers: 10 - Number of choices for main courses: 15 - Number
of choices for desserts: 8
Step 2: Use the Fundamental Principle of Counting to find the total num-
ber of different meals: Total number of different meals = Number of choices
for appetizers ×Number of choices for main courses ×Number of choices for
desserts
Total number of different meals = 10 ×15 ×8
Total number of different meals = 1200
Therefore, the customer can choose from 1200 different meals consisting of
one appetizer, one main course, and one dessert.
Question 8
Question
In how many ways can you stack 5 red, 3 blue, and 2 green books on a shelf if
books of the same color are indistinguishable from each other?
Solution
Step 1: We can think of this problem as finding the number of ways to arrange
the colors of the books on the shelf.
Step 2: First, we arrange the different colors of the books. There are 3
different colors (red, blue, green) which means there are 3! ways to arrange the
colors.
Step 3: Next, within each color, we can arrange the books. For the red
books, there are (5
5)= 1 way to arrange them since they are indistinguishable.
Similarly, there is 1 way to arrange the blue books and 1 way to arrange the
green books.
Step 4: Combining the arrangements of the colors and the arrangements
within each color, the total number of ways to stack the books is 3! ·1·1·1 = 6
ways.
Therefore, there are 6 ways to stack the books on the shelf.
Question 9
Question
In how many ways can a committee of 5 people be chosen from a group of 10
individuals where 3 of them are professors and 7 are students?
6
Solution
Step 1: Determine the number of ways to choose 3 professors. Since there are
3 professors out of 10 individuals, we can choose 3 professors in (3
3)= 1 way.
Step 2: Determine the number of ways to choose 2 students. Since there are
7 students out of 10 individuals, we can choose 2 students in (7
2)= 21 ways.
Step 3: Multiply the number of ways from Step 1 and Step 2 to find the
total number of ways to choose a committee of 5 people. 1×21 = 21
Therefore, there are 21 ways to choose a committee of 5 people from a group
of 10 individuals where 3 are professors and 7 are students.
Question 10
Question
In a group of 10 students, how many ways can we select 4 students to form a
study group?
Solution
To solve this problem, we will use the concept of combinations (a type of com-
binatorial analysis) which helps us determine the number of ways to choose a
subset from a larger set without considering the order of selection.
Step 1: Recall that the number of ways to choose kobjects from a set of n
objects is given by the formula for combinations:
C(n, k) = n!
k!(n−k)!
Step 2: In this case, we have a group of 10 students from which we want
to select a study group of 4 students. So, we have n= 10 (total number of
students) and k= 4 (number of students to be selected).
Step 3: Now, plug in the values of nand kinto the combination formula
to find the number of ways to select a study group of 4 students from a group
of 10 students:
C(10,4) = 10!
4!(10 −4)!
Step 4: Calculate the factorials:
10! = 10 ×9×8×7×6×5×4×3×2×1
4! = 4 ×3×2×1
6! = 6 ×5×4×3×2×1
Step 5: Substitute the factorials back into the combination formula:
C(10,4) = 10 ×9×8×7×6×5×4×3×2×1
(4 ×3×2×1) ×(6 ×5×4×3×2×1)
7
Step 6: Simplify the expression:
C(10,4) = 10 ×9×8×7
4×3×2×1=5040
24 = 210
Step 7: Therefore, there are 210 ways to select 4 students from a group of
10 students to form a study group.
Question 11
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
Find the number of ways the committee can be formed if it must consist of at
least 3 men.
Solution
Step 1: Calculate the number of ways to form a committee with exactly 3 men
and 2 women. Let’s first calculate the ways to choose 3 men out of 10 and 2
women out of 8. This can be done by using the combination formula (n
r)=
n!
r!(n−r)! . So, the number of ways to choose 3 men from 10 is (10
3)=10!
3!(10−3)!
and the number of ways to choose 2 women from 8 is (8
2)=8!
2!(8−2)! .
Step 2: Calculate the total number of ways to form a committee with at
least 3 men. We can form a committee with exactly 3, 4, or 5 men. So, the
total number of ways to form a committee with at least 3 men is the sum of the
number of ways to form a committee with exactly 3, 4, or 5 men.
Step 3: Calculate the total number of ways to form a committee with at
least 3 men.
Total ways =(10
3)×(8
2)+(10
4)×(8
1)+(10
5)×(8
0)
Step 4: Perform the calculations.
(10
3)=10!
3!(10 −3)! = 120
(8
2)=8!
2!(8 −2)! = 28
(10
4)=10!
4!(10 −4)! = 210
(8
1)=8!
1!(8 −1)! = 8
(10
5)=10!
5!(10 −5)! = 252
8
(8
0)= 1
So the total number of ways to form a committee of 5 people consisting of
at least 3 men is:
120 ×28 + 210 ×8 + 252 ×1 = 3360 + 1680 + 252 = 5292
Therefore, there are 5,292 ways to form the committee with at least 3 men.
Question 12
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
In how many ways can the committee be formed if it must consist of at least 3
women?
Solution
Step 1: Calculate the number of ways to choose exactly 3 women and 2 men for
the committee. Since we need to choose exactly 3 women from 8 women and 2
men from 10 men, we can use the combination formula to calculate this.
(8
3)×(10
2)=8!
3!5! ×10!
2!8! = 56 ×45 = 2520
Step 2: Calculate the number of ways to choose exactly 4 women and 1 man
for the committee. Now, we need to choose exactly 4 women from 8 women and
1 man from 10 men.
(8
4)×(10
1)=8!
4!4! ×10!
1!9! = 70 ×10 = 700
Step 3: Calculate the number of ways to choose all 5 committee members as
women. For this case, we need to choose all 5 women from the 8 available.
(8
5)=8!
5!3! = 56
Step 4: Add up the results from Step 1, Step 2, and Step 3 to find the total
number of ways to form the committee.
2520 + 700 + 56 = 3276
Therefore, there are 3276 ways to form a committee of 5 people consisting
of at least 3 women from a group of 10 men and 8 women.
9
Question 13
Question
A committee of 5 people is to be formed from a group of 10 students (5 male
and 5 female). Find the number of ways the committee can be formed if it must
consist of at least 2 male students.
Solution
Step 1: Calculate the number of ways to choose 2, 3, 4, or 5 male students from
the group of 5 male students.
• Choose 2 male students: (5
2)= 10 ways.
• Choose 3 male students: (5
3)= 10 ways.
• Choose 4 male students: (5
4)= 5 ways.
• Choose 5 male students: (5
5)= 1 way.
Step 2: Calculate the number of ways to choose the remaining members of
the committee (from the females).
• For each combination of male students chosen in Step 1, we need to choose
the remaining members from the group of 5 female students.
Step 3: Calculate the total number of ways to form the committee.
• Add the results from Step 1 and Step 2 to find the total number of ways
to form the committee with at least 2 male students.
Therefore, the total number of ways the committee can be formed is:
(5
2)×(5
3)+(5
3)×(5
2)+(5
4)×(5
1)+(5
5)×(5
0)
Question 14
Question
A committee of 5 people is to be formed from a group of 7 men and 5 women. If
at least 2 men must be on the committee, how many different committees can
be formed?
10
Solution
Step 1: Calculate the total number of ways to form a committee without any
restrictions. There are 12 people total, and we need to choose a committee of
5. So, the total number of ways to form a committee without any restrictions
is given by the combination formula: (12
5).
Step 2: Calculate the number of ways to form a committee with no men.
Since at least 2 men are required on the committee, we must find the number
of ways to form a committee without any men. Given that there are 5 women,
the number of ways to choose 5 women from a group of 5 is (5
5).
Step 3: Calculate the number of ways to form a committee with exactly
1 man. Now, we need to find the number of ways to form a committee with
exactly 1 man. We choose 1 man from 7 men, which can be done in (7
1)ways.
The remaining 4 members of the committee must be women, chosen from a
group of 5, which can be done in (5
4)ways.
Step 4: Calculate the number of ways to form a committee with exactly 2
men. Next, we calculate the number of ways to form a committee with exactly
2 men. We choose 2 men from 7 men, which can be done in (7
2)ways. The
remaining 3 members of the committee must be women, chosen from a group
of 5, which can be done in (5
3)ways.
Step 5: Calculate the total number of ways to form a committee with at
least 2 men. To calculate the total number of ways to form a committee with
at least 2 men, we sum the number of ways to form a committee with exactly
2 men, exactly 1 man, and no men. Therefore, the total number of ways is:
(7
2)×(5
3)+(7
1)×(5
4)+(5
5).
Step 6: Perform the calculations. (7
2)= 21,(5
3)= 10,(7
1)= 7,(5
4)= 5.
Therefore, the total number of ways to form a committee with at least 2
men is: 21 ×10 + 7 ×5 + 1 = 220 + 35 + 1 = 256.
So, there are 256 different committees that can be formed.
Question 15
Question
In a group of 10 students, how many ways can we choose a committee of 4
students with a president, a vice president, and a treasurer? Assume that no
student can hold more than one position.
Solution
Step 1: To choose the president, we have 10 options.
Step 2: After choosing the president, there are 9 students left to choose from
for the vice president position.
Step 3: After choosing the president and vice president, there are 8 students
left to choose from for the treasurer position.
11
Step 4: Finally, after assigning the positions of president, vice president, and
treasurer, there are 7 students left to fill the last remaining position.
Therefore, the total number of ways to choose the committee is given by:
10 ×9×8×7 = 5040
So, there are 5040 ways to choose a committee of 4 students with a president,
a vice president, and a treasurer from a group of 10 students.
Question 16
Question
In how many ways can a committee of 3 men and 4 women be formed from a
group of 10 men and 8 women?
Solution
Step 1: Calculate the number of ways to choose 3 men from 10.
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120
Step 2: Calculate the number of ways to choose 4 women from 8.
(8
4)=8!
4!(8 −4)! =8×7×6×5
4×3×2×1= 70
Step 3: Multiply the results from steps 1 and 2 to find the total number of
ways to form the committee.
120 ×70 = 8400
Therefore, there are 8400 ways to form a committee of 3 men and 4 women
from a group of 10 men and 8 women.
Question 17
Question
A committee of 5 people is to be formed from a group of 8 men and 6 women.
How many different committees can be formed if the committee must consist of
3 men and 2 women?
12
Solution
Step 1: Calculate the number of ways to choose 3 men from 8.
(8
3)=8!
3!(8 −3)! =8!
3!5! =8×7×6
3×2×1= 56
Step 2: Calculate the number of ways to choose 2 women from 6.
(6
2)=6!
2!(6 −2)! =6!
2!4! =6×5
2×1= 15
Step 3: Multiply the number of ways from Step 1 and Step 2 to get the total
number of committees that can be formed.
56 ×15 = 840
Therefore, there are 840 different committees that can be formed if the
committee must consist of 3 men and 2 women.
Question 18
Question
A selection committee needs to choose a president, a vice president, and a
secretary from a group of 10 candidates. How many different ways can these
three positions be filled if no person can hold more than one position?
Solution
Step 1: To determine the number of ways to choose the president (the first
position), we use the multiplication rule. Since there are 10 candidates and no
person can hold more than one position, there are 10 candidates eligible for the
position of president. Therefore, there are 10 ways to choose the president.
Step 2: After selecting the president, we move on to choose the vice president
(the second position). Since the president has already been chosen, there are 9
candidates remaining for the position of vice president. Therefore, there are 9
ways to choose the vice president.
Step 3: Finally, after selecting the president and vice president, there are 8
candidates left for the position of secretary. Thus, there are 8 ways to choose
the secretary.
Step 4: To find the total number of ways to fill all three positions, we multiply
the number of ways for each position: 10 ×9×8 = 720
Therefore, there are 720 different ways to fill the positions of president, vice
president, and secretary from a group of 10 candidates.
13
Question 19
Question
A committee of 5 people is to be formed from a group of 7 men and 6 women.
Find the probability that the committee contains at least 2 men.
Solution
Step 1: Find the total number of ways to form a committee of 5 people. Since
the committee can be made up of either all men, all women, or a mix of men
and women, we have to consider all possibilities. - All men: (7
5)ways to choose
5 men. - All women: (6
5)ways to choose 5 women. - Mix of men and women:
(7
4)×(6
1)ways to choose 4 men and 1 woman, or (7
3)×(6
2)ways to choose 3 men
and 2 women.
Therefore, the total number of ways to form a committee of 5 people is:
(7
5)+(6
5)+((7
4)×(6
1))+((7
3)×(6
2))
Step 2: Find the number of ways to form a committee with fewer than 2
men (0 or 1 man). - 0 men: (6
5)ways to choose 5 women. - 1 man: (7
1)×(6
4)
ways to choose 1 man and 4 women.
Therefore, the number of ways to form a committee with fewer than 2 men
is: (6
5)+((7
1)×(6
4))
Step 3: Calculate the probability of forming a committee with at least 2
men. The probability is given by:
1−Number of ways to form a committee with fewer than 2 men
Total number of ways to form a committee
Now, you can plug in the values to find the probability.
Question 20
Question
A group of 10 students are taking a multiple-choice quiz with 5 questions, each
having 4 possible answers. How many ways are there for the group of students
to respond to the quiz if each student responds to all 5 questions?
Solution
Step 1: For each question, there are 4 possible answers. Step 2: Since each of
the 10 students responds to all 5 questions, there are 45ways for each student
14
to respond to the quiz. Step 3: To find the total number of ways for the group
of 10 students to respond to the quiz, we multiply the number of ways for each
student to respond to the quiz by the number of students: 45×10. Step 4:
Calculating 45gives 1024, so the total number of ways for the group of 10
students to respond to the quiz is 1024 ×10 = 10240. Therefore, there are
10,240 ways for the group of students to respond to the quiz.
Question 21
Question
In a group of 10 people, how many ways are there to select a committee of 4
people without regard to order?
Solution
To find the number of ways to select a committee of 4 people from a group of
10 people without regards to order, we will use the combination formula.
Step 1: Determine the number of ways to choose 4 people from 10 without
regard to order. This is denoted by (10
4).
Step 2: Calculate (10
4)using the formula:
(n
k)=n!
k!(n−k)!
where nis the total number of people (10 in this case) and kis the number of
people to select (4 in this case).
Step 3: Substitute n= 10 and k= 4 into the formula:
(10
4)=10!
4!(10 −4)! =10!
4!6!
Step 4: Simplify the expression:
(10
4)=10 ×9×8×7×6!
(4 ×3×2×1) ×6! =10 ×9×8×7
4×3×2×1= 210
Therefore, there are 210 ways to select a committee of 4 people from a group
of 10 people without regard to order.
Question 22
Question
In a committee of 8 people, how many ways can we choose a president, vice-
president, and treasurer if no person can hold more than one office?
15
Solution
Step 1: Determine the number of ways to choose the president. Since there are
8 people in the committee, there are 8 choices for the president.
Step 2: After choosing the president, there are 7 people left to choose from
for the vice-president. Therefore, there are 7 choices for the vice-president.
Step 3: Finally, after choosing the president and vice-president, there are 6
people remaining for the position of treasurer. Thus, there are 6 choices for the
treasurer.
Step 4: To find the total number of ways to choose a president, vice-
president, and treasurer without anyone holding more than one office, we mul-
tiply the number of choices at each step: 8×7×6
Therefore, there are 8×7×6 = 336 ways to choose a president, vice-president,
and treasurer in a committee of 8 people where no person can hold more than
one office.
Question 23
Question
In a group of 10 students, how many ways can you choose a committee of 4
students if two particular students refuse to serve on the committee together?
Solution
Step 1: First, we find the total number of ways to choose a committee of 4
students from 10 students, which includes the two particular students who refuse
to serve together.
Step 2: This can be done using the combination formula (n
r)=n!
r!(n−r)! ,
where nis the total number of students and ris the number of students we
want to choose.
Step 3: Substituting n= 10 and r= 4 into the formula, we get: (10
4)=
10!
4!(10−4)! =10×9×8×7
4×3×2×1= 210 ways.
Step 4: Next, we find the number of ways to form a committee of 4 members
without the two particular students who refuse to serve together.
Step 5: This can be done by subtracting the number of ways to choose
a committee of 4 students from the group of 8 students (excluding the two
particular students) from the total number of ways found in Step 3.
Step 6: Using the combination formula with n= 8 and r= 4, we get:
(8
4)=8!
4!(8−4)! =8×7×6×5
4×3×2×1= 70 ways.
Step 7: Therefore, the number of ways to choose a committee of 4 students
from the group of 10 students with the condition that the two particular students
refuse to serve together is 210 −70 = 140 ways.
16
Question 24
Question
In a group of 10 people, how many ways are there to select a committee of 4
people with a president, a treasurer, and 2 regular members?
Solution
Step 1: To determine the number of ways to choose the president, we select one
person from the 10 people. There are 10 ways to choose the president.
Step 2: After selecting the president, we need to choose the treasurer. Since
the president has already been chosen, there are 9 remaining people to choose
from. There are 9 ways to choose the treasurer.
Step 3: For the 2 regular members, we need to choose 2 people from the
remaining 8 people (after selecting the president and the treasurer). We can
calculate this using combinations. The number of ways to choose 2 regular
members from 8 people is given by (8
2)=8!
2!(8−2)! = 28 ways.
Step 4: To find the total number of ways to select the committee of 4 people
with a president, a treasurer, and 2 regular members, we multiply the number
of ways for each position: 10 ×9×28 = 2520.
Therefore, there are 2520 ways to select a committee of 4 people with a
president, a treasurer, and 2 regular members from a group of 10 people.
Question 25
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 possible
committees can be formed?
Solution
Step 1: Calculate the number of ways to choose 2 men from the 10 men. There
are (10
2)ways to choose 2 men.
Step 2: Calculate the number of ways to choose 2 women from the 8 women.
There are (8
2)ways to choose 2 women.
Step 3: Calculate the number of ways to choose the remaining person (either
man or woman) for the committee. Since the committee must consist of at least
2 men and 2 women, we can either choose 3 men and 2 women or 2 men and 3
women for the committee.
For the first case: - Choose 3 men from the remaining 8 men: (8
3)ways. -
Choose 2 women from the remaining 8 women: (8
2)ways.
For the second case: - Choose 2 men from the remaining 8 men: (8
2)ways.
- Choose 3 women from the remaining 8 women: (8
3)ways.
17
Step 4: Calculate the total number of ways to form the committee. The
total number of ways to form the committee is the sum of the ways obtained
from the two cases: (10
2)×(8
2)×(8
3)+(10
2)×(8
2)×(8
3)
Simplify to find the total number of possible committees.
18
(10
5)=30240
120
Step 6: Finally, simplifying the fraction gives us
(10
5)= 252
Therefore, there are 252 ways to select 5 books from a shelf containing 10
different books when the order of selection doesn’t matter.
Question 2
Question
In a group of 12 people, how many ways can we form a committee of 4 people
where 2 of them are designated as co-chairs (i.e., order matters)?
Solution
Step 1: To find the number of ways to choose 4 people from 12 to form a
committee, we will use the combination formula (n
k)=n!
k!(n−k)! . In this case,
n= 12 and k= 4.
Number of ways to choose 4 people from 12 =(12
4)=12!
4!(12 −4)!
Step 2: Calculate the number of ways to choose 4 people from 12.
(12
4)=12!
4!(12 −4)! =12 ×11 ×10 ×9
4×3×2×1
Step 3: Simplify the expression to find the total number of ways to choose
4 people from 12.
(12
4)= 495
Step 4: For each committee of 4 people, there are 2 co-chairs to be selected.
Since order matters for the co-chairs, we can select the first co-chair in 4 ways
(since there are 4 committee members) and the second co-chair in 3 ways. The
total number of ways to select 2 co-chairs from the 4 committee members is
4×3 = 12.
Step 5: Finally, multiply the number of ways to form a committee of 4 people
with the number of ways to choose 2 co-chairs.
T otal number of ways =Number of ways to form a committee×Number of ways to choose 2 co-chairs
T otal number of ways = 495 ×12
T otal number of ways = 5940
Therefore, there are 5940 ways to form a committee of 4 people with 2
designated as co-chairs from a group of 12 people.
2
Question 3
Question
In a group of 10 people, how many ways can we select a committee of 5 people
if 2 specific people, A and B, refuse to serve on the committee together?
Solution
To find the number of ways to select a committee of 5 people from a group of 10
people where A and B refuse to serve together, we need to consider two cases:
when A is on the committee and when B is on the committee.
Case 1: A is on the committee
• Select A to be on the committee: (1
1)
• Select 3 more people from the remaining 8 (excluding B): (8
3)
There are (1
1)×(8
3)= 56 ways for A to be on the committee.
Case 2: B is on the committee
• Select B to be on the committee: (1
1)
• Select 3 more people from the remaining 8 (excluding A): (8
3)
There are (1
1)×(8
3)= 56 ways for B to be on the committee.
Therefore, the total number of ways to select a committee of 5 people with
A and B not serving together is 56 + 56 = 112.
Question 4
Question
A committee of 5 people is to be formed from a group of 8 women and 6 men.
In how many ways can the committee be formed if there must be at least 3
women on it?
Solution
Step 1: Calculate the number of ways to form a committee with exactly 3 women
and 2 men. There are (8
3)ways to choose 3 women from 8 and (6
2)ways to choose
2 men from 6. Therefore, the number of ways to form a committee with exactly
3 women and 2 men is:
(8
3)×(6
2)= 56 ×15 = 840
Step 2: Calculate the number of ways to form a committee with exactly 4
women and 1 man. There are (8
4)ways to choose 4 women from 8 and (6
1)ways
3
to choose 1 man from 6. Therefore, the number of ways to form a committee
with exactly 4 women and 1 man is:
(8
4)×(6
1)= 70 ×6 = 420
Step 3: Calculate the number of ways to form a committee with 5 women.
There are (8
5)ways to choose 5 women from 8 (since all women have to be on
the committee). Therefore, the number of ways to form a committee with 5
women is: (8
5)= 56
Step 4: Add the results from Step 1, Step 2, and Step 3 to find the total
number of ways to form the committee. Total number of ways = 840 + 420 +
56 = 1316
Therefore, the committee can be formed in 1316 different ways if there must
be at least 3 women on it.
Question 5
Question
In a group of 10 people, how many ways are there to choose a committee of
3 people if 2 specific people (Alice and Bob) refuse to be on the committee
together?
Solution
Step 1: The total number of ways to choose a committee of 3 people out of 10 is
given by the combination formula (n
r)=n!
r!(n−r)! . Thus, the total number of ways
to choose a committee of 3 people from 10 is (10
3)=10!
3!(10−3)! =10×9×8
3×2×1= 120.
Step 2: Now, we need to subtract the number of ways to choose a committee
that includes both Alice and Bob. If Alice and Bob are both on the committee,
we need to choose 1 more person from the remaining 8 people. The number of
ways to do this is (8
1)= 8.
Step 3: Finally, we need to subtract the number of ways to choose a commit-
tee that includes both Alice and Bob from the total number of ways to choose
a committee. We have 120 −8 = 112.
Therefore, there are 112 ways to choose a committee of 3 people from a
group of 10 people if Alice and Bob refuse to be on the committee together.
4
Question 6
Question
In a group of 10 people, how many ways can we choose a committee of 4 people
if two particular individuals, Alex and Beth, refuse to serve on the committee
together?
Solution
Step 1: First, we find the total number of ways to choose a committee of 4
people from the group of 10 without any restrictions. This can be calculated
using the combination formula:
(10
4)=10!
4!(10 −4)! =10 ×9×8×7
4×3×2×1= 210.
Step 2: Next, we find the number of ways to choose a committee of 4 people
where Alex and Beth are both members. This can be calculated by choosing 2
people from the remaining 8 (excluding Alex and Beth) and adding Alex and
Beth. This can be done as follows:
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28.
However, we need to take into account that Alex and Beth cannot serve on the
committee together, so we subtract this value from the total.
Step 3: Therefore, the number of ways to choose a committee of 4 people
where Alex and Beth refuse to serve together is:
210 −28 = 182.
Hence, there are 182 ways to choose a committee of 4 people from a group
of 10 people where Alex and Beth refuse to serve on the committee together.
Question 7
Question
A restaurant offers a menu consisting of 10 appetizers, 15 main courses, and
8 desserts. A customer wants to choose a meal consisting of one appetizer,
one main course, and one dessert. How many different meals can the customer
choose from?
Solution
Let’s use the Fundamental Principle of Counting to determine the total number
of different meals the customer can choose.
5
Step 1: Determine the number of choices for each category: - Number of
choices for appetizers: 10 - Number of choices for main courses: 15 - Number
of choices for desserts: 8
Step 2: Use the Fundamental Principle of Counting to find the total num-
ber of different meals: Total number of different meals = Number of choices
for appetizers ×Number of choices for main courses ×Number of choices for
desserts
Total number of different meals = 10 ×15 ×8
Total number of different meals = 1200
Therefore, the customer can choose from 1200 different meals consisting of
one appetizer, one main course, and one dessert.
Question 8
Question
In how many ways can you stack 5 red, 3 blue, and 2 green books on a shelf if
books of the same color are indistinguishable from each other?
Solution
Step 1: We can think of this problem as finding the number of ways to arrange
the colors of the books on the shelf.
Step 2: First, we arrange the different colors of the books. There are 3
different colors (red, blue, green) which means there are 3! ways to arrange the
colors.
Step 3: Next, within each color, we can arrange the books. For the red
books, there are (5
5)= 1 way to arrange them since they are indistinguishable.
Similarly, there is 1 way to arrange the blue books and 1 way to arrange the
green books.
Step 4: Combining the arrangements of the colors and the arrangements
within each color, the total number of ways to stack the books is 3! ·1·1·1 = 6
ways.
Therefore, there are 6 ways to stack the books on the shelf.
Question 9
Question
In how many ways can a committee of 5 people be chosen from a group of 10
individuals where 3 of them are professors and 7 are students?
6
Solution
Step 1: Determine the number of ways to choose 3 professors. Since there are
3 professors out of 10 individuals, we can choose 3 professors in (3
3)= 1 way.
Step 2: Determine the number of ways to choose 2 students. Since there are
7 students out of 10 individuals, we can choose 2 students in (7
2)= 21 ways.
Step 3: Multiply the number of ways from Step 1 and Step 2 to find the
total number of ways to choose a committee of 5 people. 1×21 = 21
Therefore, there are 21 ways to choose a committee of 5 people from a group
of 10 individuals where 3 are professors and 7 are students.
Question 10
Question
In a group of 10 students, how many ways can we select 4 students to form a
study group?
Solution
To solve this problem, we will use the concept of combinations (a type of com-
binatorial analysis) which helps us determine the number of ways to choose a
subset from a larger set without considering the order of selection.
Step 1: Recall that the number of ways to choose kobjects from a set of n
objects is given by the formula for combinations:
C(n, k) = n!
k!(n−k)!
Step 2: In this case, we have a group of 10 students from which we want
to select a study group of 4 students. So, we have n= 10 (total number of
students) and k= 4 (number of students to be selected).
Step 3: Now, plug in the values of nand kinto the combination formula
to find the number of ways to select a study group of 4 students from a group
of 10 students:
C(10,4) = 10!
4!(10 −4)!
Step 4: Calculate the factorials:
10! = 10 ×9×8×7×6×5×4×3×2×1
4! = 4 ×3×2×1
6! = 6 ×5×4×3×2×1
Step 5: Substitute the factorials back into the combination formula:
C(10,4) = 10 ×9×8×7×6×5×4×3×2×1
(4 ×3×2×1) ×(6 ×5×4×3×2×1)
7
Step 6: Simplify the expression:
C(10,4) = 10 ×9×8×7
4×3×2×1=5040
24 = 210
Step 7: Therefore, there are 210 ways to select 4 students from a group of
10 students to form a study group.
Question 11
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
Find the number of ways the committee can be formed if it must consist of at
least 3 men.
Solution
Step 1: Calculate the number of ways to form a committee with exactly 3 men
and 2 women. Let’s first calculate the ways to choose 3 men out of 10 and 2
women out of 8. This can be done by using the combination formula (n
r)=
n!
r!(n−r)! . So, the number of ways to choose 3 men from 10 is (10
3)=10!
3!(10−3)!
and the number of ways to choose 2 women from 8 is (8
2)=8!
2!(8−2)! .
Step 2: Calculate the total number of ways to form a committee with at
least 3 men. We can form a committee with exactly 3, 4, or 5 men. So, the
total number of ways to form a committee with at least 3 men is the sum of the
number of ways to form a committee with exactly 3, 4, or 5 men.
Step 3: Calculate the total number of ways to form a committee with at
least 3 men.
Total ways =(10
3)×(8
2)+(10
4)×(8
1)+(10
5)×(8
0)
Step 4: Perform the calculations.
(10
3)=10!
3!(10 −3)! = 120
(8
2)=8!
2!(8 −2)! = 28
(10
4)=10!
4!(10 −4)! = 210
(8
1)=8!
1!(8 −1)! = 8
(10
5)=10!
5!(10 −5)! = 252
8
(8
0)= 1
So the total number of ways to form a committee of 5 people consisting of
at least 3 men is:
120 ×28 + 210 ×8 + 252 ×1 = 3360 + 1680 + 252 = 5292
Therefore, there are 5,292 ways to form the committee with at least 3 men.
Question 12
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
In how many ways can the committee be formed if it must consist of at least 3
women?
Solution
Step 1: Calculate the number of ways to choose exactly 3 women and 2 men for
the committee. Since we need to choose exactly 3 women from 8 women and 2
men from 10 men, we can use the combination formula to calculate this.
(8
3)×(10
2)=8!
3!5! ×10!
2!8! = 56 ×45 = 2520
Step 2: Calculate the number of ways to choose exactly 4 women and 1 man
for the committee. Now, we need to choose exactly 4 women from 8 women and
1 man from 10 men.
(8
4)×(10
1)=8!
4!4! ×10!
1!9! = 70 ×10 = 700
Step 3: Calculate the number of ways to choose all 5 committee members as
women. For this case, we need to choose all 5 women from the 8 available.
(8
5)=8!
5!3! = 56
Step 4: Add up the results from Step 1, Step 2, and Step 3 to find the total
number of ways to form the committee.
2520 + 700 + 56 = 3276
Therefore, there are 3276 ways to form a committee of 5 people consisting
of at least 3 women from a group of 10 men and 8 women.
9
Question 13
Question
A committee of 5 people is to be formed from a group of 10 students (5 male
and 5 female). Find the number of ways the committee can be formed if it must
consist of at least 2 male students.
Solution
Step 1: Calculate the number of ways to choose 2, 3, 4, or 5 male students from
the group of 5 male students.
• Choose 2 male students: (5
2)= 10 ways.
• Choose 3 male students: (5
3)= 10 ways.
• Choose 4 male students: (5
4)= 5 ways.
• Choose 5 male students: (5
5)= 1 way.
Step 2: Calculate the number of ways to choose the remaining members of
the committee (from the females).
• For each combination of male students chosen in Step 1, we need to choose
the remaining members from the group of 5 female students.
Step 3: Calculate the total number of ways to form the committee.
• Add the results from Step 1 and Step 2 to find the total number of ways
to form the committee with at least 2 male students.
Therefore, the total number of ways the committee can be formed is:
(5
2)×(5
3)+(5
3)×(5
2)+(5
4)×(5
1)+(5
5)×(5
0)
Question 14
Question
A committee of 5 people is to be formed from a group of 7 men and 5 women. If
at least 2 men must be on the committee, how many different committees can
be formed?
10
Solution
Step 1: Calculate the total number of ways to form a committee without any
restrictions. There are 12 people total, and we need to choose a committee of
5. So, the total number of ways to form a committee without any restrictions
is given by the combination formula: (12
5).
Step 2: Calculate the number of ways to form a committee with no men.
Since at least 2 men are required on the committee, we must find the number
of ways to form a committee without any men. Given that there are 5 women,
the number of ways to choose 5 women from a group of 5 is (5
5).
Step 3: Calculate the number of ways to form a committee with exactly
1 man. Now, we need to find the number of ways to form a committee with
exactly 1 man. We choose 1 man from 7 men, which can be done in (7
1)ways.
The remaining 4 members of the committee must be women, chosen from a
group of 5, which can be done in (5
4)ways.
Step 4: Calculate the number of ways to form a committee with exactly 2
men. Next, we calculate the number of ways to form a committee with exactly
2 men. We choose 2 men from 7 men, which can be done in (7
2)ways. The
remaining 3 members of the committee must be women, chosen from a group
of 5, which can be done in (5
3)ways.
Step 5: Calculate the total number of ways to form a committee with at
least 2 men. To calculate the total number of ways to form a committee with
at least 2 men, we sum the number of ways to form a committee with exactly
2 men, exactly 1 man, and no men. Therefore, the total number of ways is:
(7
2)×(5
3)+(7
1)×(5
4)+(5
5).
Step 6: Perform the calculations. (7
2)= 21,(5
3)= 10,(7
1)= 7,(5
4)= 5.
Therefore, the total number of ways to form a committee with at least 2
men is: 21 ×10 + 7 ×5 + 1 = 220 + 35 + 1 = 256.
So, there are 256 different committees that can be formed.
Question 15
Question
In a group of 10 students, how many ways can we choose a committee of 4
students with a president, a vice president, and a treasurer? Assume that no
student can hold more than one position.
Solution
Step 1: To choose the president, we have 10 options.
Step 2: After choosing the president, there are 9 students left to choose from
for the vice president position.
Step 3: After choosing the president and vice president, there are 8 students
left to choose from for the treasurer position.
11
Step 4: Finally, after assigning the positions of president, vice president, and
treasurer, there are 7 students left to fill the last remaining position.
Therefore, the total number of ways to choose the committee is given by:
10 ×9×8×7 = 5040
So, there are 5040 ways to choose a committee of 4 students with a president,
a vice president, and a treasurer from a group of 10 students.
Question 16
Question
In how many ways can a committee of 3 men and 4 women be formed from a
group of 10 men and 8 women?
Solution
Step 1: Calculate the number of ways to choose 3 men from 10.
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120
Step 2: Calculate the number of ways to choose 4 women from 8.
(8
4)=8!
4!(8 −4)! =8×7×6×5
4×3×2×1= 70
Step 3: Multiply the results from steps 1 and 2 to find the total number of
ways to form the committee.
120 ×70 = 8400
Therefore, there are 8400 ways to form a committee of 3 men and 4 women
from a group of 10 men and 8 women.
Question 17
Question
A committee of 5 people is to be formed from a group of 8 men and 6 women.
How many different committees can be formed if the committee must consist of
3 men and 2 women?
12
Solution
Step 1: Calculate the number of ways to choose 3 men from 8.
(8
3)=8!
3!(8 −3)! =8!
3!5! =8×7×6
3×2×1= 56
Step 2: Calculate the number of ways to choose 2 women from 6.
(6
2)=6!
2!(6 −2)! =6!
2!4! =6×5
2×1= 15
Step 3: Multiply the number of ways from Step 1 and Step 2 to get the total
number of committees that can be formed.
56 ×15 = 840
Therefore, there are 840 different committees that can be formed if the
committee must consist of 3 men and 2 women.
Question 18
Question
A selection committee needs to choose a president, a vice president, and a
secretary from a group of 10 candidates. How many different ways can these
three positions be filled if no person can hold more than one position?
Solution
Step 1: To determine the number of ways to choose the president (the first
position), we use the multiplication rule. Since there are 10 candidates and no
person can hold more than one position, there are 10 candidates eligible for the
position of president. Therefore, there are 10 ways to choose the president.
Step 2: After selecting the president, we move on to choose the vice president
(the second position). Since the president has already been chosen, there are 9
candidates remaining for the position of vice president. Therefore, there are 9
ways to choose the vice president.
Step 3: Finally, after selecting the president and vice president, there are 8
candidates left for the position of secretary. Thus, there are 8 ways to choose
the secretary.
Step 4: To find the total number of ways to fill all three positions, we multiply
the number of ways for each position: 10 ×9×8 = 720
Therefore, there are 720 different ways to fill the positions of president, vice
president, and secretary from a group of 10 candidates.
13
Question 19
Question
A committee of 5 people is to be formed from a group of 7 men and 6 women.
Find the probability that the committee contains at least 2 men.
Solution
Step 1: Find the total number of ways to form a committee of 5 people. Since
the committee can be made up of either all men, all women, or a mix of men
and women, we have to consider all possibilities. - All men: (7
5)ways to choose
5 men. - All women: (6
5)ways to choose 5 women. - Mix of men and women:
(7
4)×(6
1)ways to choose 4 men and 1 woman, or (7
3)×(6
2)ways to choose 3 men
and 2 women.
Therefore, the total number of ways to form a committee of 5 people is:
(7
5)+(6
5)+((7
4)×(6
1))+((7
3)×(6
2))
Step 2: Find the number of ways to form a committee with fewer than 2
men (0 or 1 man). - 0 men: (6
5)ways to choose 5 women. - 1 man: (7
1)×(6
4)
ways to choose 1 man and 4 women.
Therefore, the number of ways to form a committee with fewer than 2 men
is: (6
5)+((7
1)×(6
4))
Step 3: Calculate the probability of forming a committee with at least 2
men. The probability is given by:
1−Number of ways to form a committee with fewer than 2 men
Total number of ways to form a committee
Now, you can plug in the values to find the probability.
Question 20
Question
A group of 10 students are taking a multiple-choice quiz with 5 questions, each
having 4 possible answers. How many ways are there for the group of students
to respond to the quiz if each student responds to all 5 questions?
Solution
Step 1: For each question, there are 4 possible answers. Step 2: Since each of
the 10 students responds to all 5 questions, there are 45ways for each student
14
to respond to the quiz. Step 3: To find the total number of ways for the group
of 10 students to respond to the quiz, we multiply the number of ways for each
student to respond to the quiz by the number of students: 45×10. Step 4:
Calculating 45gives 1024, so the total number of ways for the group of 10
students to respond to the quiz is 1024 ×10 = 10240. Therefore, there are
10,240 ways for the group of students to respond to the quiz.
Question 21
Question
In a group of 10 people, how many ways are there to select a committee of 4
people without regard to order?
Solution
To find the number of ways to select a committee of 4 people from a group of
10 people without regards to order, we will use the combination formula.
Step 1: Determine the number of ways to choose 4 people from 10 without
regard to order. This is denoted by (10
4).
Step 2: Calculate (10
4)using the formula:
(n
k)=n!
k!(n−k)!
where nis the total number of people (10 in this case) and kis the number of
people to select (4 in this case).
Step 3: Substitute n= 10 and k= 4 into the formula:
(10
4)=10!
4!(10 −4)! =10!
4!6!
Step 4: Simplify the expression:
(10
4)=10 ×9×8×7×6!
(4 ×3×2×1) ×6! =10 ×9×8×7
4×3×2×1= 210
Therefore, there are 210 ways to select a committee of 4 people from a group
of 10 people without regard to order.
Question 22
Question
In a committee of 8 people, how many ways can we choose a president, vice-
president, and treasurer if no person can hold more than one office?
15
Solution
Step 1: Determine the number of ways to choose the president. Since there are
8 people in the committee, there are 8 choices for the president.
Step 2: After choosing the president, there are 7 people left to choose from
for the vice-president. Therefore, there are 7 choices for the vice-president.
Step 3: Finally, after choosing the president and vice-president, there are 6
people remaining for the position of treasurer. Thus, there are 6 choices for the
treasurer.
Step 4: To find the total number of ways to choose a president, vice-
president, and treasurer without anyone holding more than one office, we mul-
tiply the number of choices at each step: 8×7×6
Therefore, there are 8×7×6 = 336 ways to choose a president, vice-president,
and treasurer in a committee of 8 people where no person can hold more than
one office.
Question 23
Question
In a group of 10 students, how many ways can you choose a committee of 4
students if two particular students refuse to serve on the committee together?
Solution
Step 1: First, we find the total number of ways to choose a committee of 4
students from 10 students, which includes the two particular students who refuse
to serve together.
Step 2: This can be done using the combination formula (n
r)=n!
r!(n−r)! ,
where nis the total number of students and ris the number of students we
want to choose.
Step 3: Substituting n= 10 and r= 4 into the formula, we get: (10
4)=
10!
4!(10−4)! =10×9×8×7
4×3×2×1= 210 ways.
Step 4: Next, we find the number of ways to form a committee of 4 members
without the two particular students who refuse to serve together.
Step 5: This can be done by subtracting the number of ways to choose
a committee of 4 students from the group of 8 students (excluding the two
particular students) from the total number of ways found in Step 3.
Step 6: Using the combination formula with n= 8 and r= 4, we get:
(8
4)=8!
4!(8−4)! =8×7×6×5
4×3×2×1= 70 ways.
Step 7: Therefore, the number of ways to choose a committee of 4 students
from the group of 10 students with the condition that the two particular students
refuse to serve together is 210 −70 = 140 ways.
16
Question 24
Question
In a group of 10 people, how many ways are there to select a committee of 4
people with a president, a treasurer, and 2 regular members?
Solution
Step 1: To determine the number of ways to choose the president, we select one
person from the 10 people. There are 10 ways to choose the president.
Step 2: After selecting the president, we need to choose the treasurer. Since
the president has already been chosen, there are 9 remaining people to choose
from. There are 9 ways to choose the treasurer.
Step 3: For the 2 regular members, we need to choose 2 people from the
remaining 8 people (after selecting the president and the treasurer). We can
calculate this using combinations. The number of ways to choose 2 regular
members from 8 people is given by (8
2)=8!
2!(8−2)! = 28 ways.
Step 4: To find the total number of ways to select the committee of 4 people
with a president, a treasurer, and 2 regular members, we multiply the number
of ways for each position: 10 ×9×28 = 2520.
Therefore, there are 2520 ways to select a committee of 4 people with a
president, a treasurer, and 2 regular members from a group of 10 people.
Question 25
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 possible
committees can be formed?
Solution
Step 1: Calculate the number of ways to choose 2 men from the 10 men. There
are (10
2)ways to choose 2 men.
Step 2: Calculate the number of ways to choose 2 women from the 8 women.
There are (8
2)ways to choose 2 women.
Step 3: Calculate the number of ways to choose the remaining person (either
man or woman) for the committee. Since the committee must consist of at least
2 men and 2 women, we can either choose 3 men and 2 women or 2 men and 3
women for the committee.
For the first case: - Choose 3 men from the remaining 8 men: (8
3)ways. -
Choose 2 women from the remaining 8 women: (8
2)ways.
For the second case: - Choose 2 men from the remaining 8 men: (8
2)ways.
- Choose 3 women from the remaining 8 women: (8
3)ways.
17
Step 4: Calculate the total number of ways to form the committee. The
total number of ways to form the committee is the sum of the ways obtained
from the two cases: (10
2)×(8
2)×(8
3)+(10
2)×(8
2)×(8
3)
Simplify to find the total number of possible committees.
18
