MATH 201 - INTRODUCTION TO
PROBABILITY AND STATISTICS -
Combinatorial Analysis
Question Bank - Set 6
Liberty University
Question 1
Question
A committee of 5 people is to be formed from a group of 10 individuals. If 3
of the 10 individuals are men and the rest are women, what is the probability
that the committee consists of exactly 3 men and 2 women?
Solution
Step 1: Find the total number of ways to form a committee of 5 people from a
group of 10 individuals. The total number of ways to choose 5 people from 10
is given by the combination formula:
(10
5)=10!
5!(10 −5)! =10 ×9×8×7×6
5×4×3×2×1= 252
Step 2: Determine the number of ways to select 3 men from the 3 available
men. The number of ways to choose 3 men from a group of 3 men is 1:
(3
3)= 1
Step 3: Determine the number of ways to select 2 women from the 7 available
women. The number of ways to choose 2 women from a group of 7 women is
given by the combination formula:
(7
2)=7!
2!(7 −2)! =7×6
2×1= 21
Step 4: Calculate the total number of ways to form a committee with 3 men
and 2 women. The total number of ways is given by the product of the number
of ways to choose 3 men and 2 women:
1×21 = 21
Step 5: Determine the probability of selecting a committee with 3 men and
2 women. The probability is the ratio of the number of favorable outcomes to
the total number of outcomes:
P(3 men, 2 women) = 21
252 =1
12
Therefore, the probability that the committee consists of exactly 3 men and
2 women is 1
12 .
Question 2
Question
A committee of 5 people is to be formed from a group of 8 men and 5 women.
If at least 2 men must be on the committee, how many different ways can the
committee be formed?
Solution
Step 1: Find the total number of ways to form a committee with 5 people.
Given that there are 8 men and 5 women, the total number of ways to form a
committee of 5 people from this group is given by the combination formula:
(8+5
5)=(13
5)=13!
5!(13 −5)! = 1287
Step 2: Find the number of ways to form a committee with no men. Since
at least 2 men must be on the committee, find the number of ways to form a
committee with no men first. This will be a committee of 5 women, which is
given by the combination formula:
(5
5)= 1
Step 3: Find the number of ways to form a committee with 1 man. Similarly,
find the number of ways to form a committee with 1 man and 4 women, using
the combination formula:
(8
1)×(5
4)= 8 ×5 = 40
Step 4: Calculate the total number of ways to form a committee with at
least 2 men. To find the number of ways to form a committee with at least 2
2
men, subtract the number of ways with no men and the number of ways with 1
man from the total number of ways:
1287 −1−40 = 1246
Therefore, there are 1246 different ways to form the committee with at least
2 men.
Question 3
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
How many different committees can be formed if the committee must consist of
at least 3 women?
Solution
Step 1: First, we’ll consider the number of ways to choose a committee with
exactly 3 women. Step 2: Select 3 women from the group of 8 women: (8
3)= 56
ways. Step 3: Select the remaining 2 people from the group of 10 men: (10
2)= 45
ways. Step 4: The total number of ways to form a committee with exactly 3
women is 56 ×45 = 2520 ways.
Step 5: Next, we’ll consider the number of ways to choose a committee with
exactly 4 women. Step 6: Select 4 women from the group of 8 women: (8
4)= 70
ways. Step 7: Select the remaining 1 person from the group of 10 men: (10
1)= 10
ways. Step 8: The total number of ways to form a committee with exactly 4
women is 70 ×10 = 700 ways.
Step 9: Finally, we’ll consider the number of ways to choose a committee
with all 5 women. Step 10: Select all 5 women from the group of 8 women:
(8
5)= 56 ways. Step 11: Since there are no men in this case, the remaining 0
people are automatically assigned. Step 12: The total number of ways to form
a committee with all 5 women is 56 ×1 = 56 ways.
Step 13: Therefore, the total number of different committees that can be
formed if the committee must consist of at least 3 women is 2520 + 700 + 56 =
3276 ways.
Question 4
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
How many ways can the committee be formed if it must include at least 3
women?
3
Solution
Step 1: Calculate the number of ways to choose 3 women from the 8 available.
There are (8
3)=8!
3!(8−3)! = 56 ways to choose 3 women from the group.
Step 2: Calculate the number of ways to choose the remaining 2 people from
the remaining pool of 10 men and 5 women (after selecting 3 women). There
are (10+5
2)=(15
2)=15!
2!(15−2)! = 105 ways to choose 2 people from this group.
Step 3: Calculate the total number of ways to form the committee with at
least 3 women. Multiply the results from Step 1 and Step 2 to get the total
number of ways to form the committee: 56 ×105 = 5880 ways.
Therefore, there are 5880 ways to form the committee with at least 3 women
from the group of 10 men and 8 women.
Question 5
Question
A committee of 5 people is to be formed from a group of 8 men and 6 women. If
the committee must consist of at least 2 men and 2 women, how many different
committees can be formed?
Solution
Step 1: Calculate the number of ways to choose 2 men out of 8 men. There are
(8
2)ways to choose 2 men from 8 men. This can be calculated as:
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28
Step 2: Calculate the number of ways to choose 2 women out of 6 women.
There are (6
2)ways to choose 2 women from 6 women. This can be calculated
as: (6
2)=6!
2!(6 −2)! =6×5
2×1= 15
Step 3: Calculate the number of ways to choose the remaining person (either
man or woman). There are 4 remaining people to choose from (6 men and 4
women). Therefore, there are 4 ways to choose the remaining person.
Step 4: Calculate the total number of committees that can be formed. The
total number of committees that can be formed is the product of the number
of ways to choose 2 men, 2 women, and the remaining person. Hence, the total
number of committees is:
28 ×15 ×4 = 1680
Therefore, there are 1680 different committees that can be formed with at
least 2 men and 2 women from the given group of people.
4
Question 6
Question
In a group of 10 people, how many ways can we choose a committee of 4 people
if 2 specific people must be included in the committee?
Solution
Step 1: Choose the 2 specific people who must be included in the committee.
There are (10
2)=10!
2!(10−2)! = 45 ways to choose the 2 specific people.
Step 2: Choose the remaining 2 people to complete the committee. After
selecting the 2 specific people, there are 8 people left to choose from. So, there
are (8
2)=8!
2!(8−2)! = 28 ways to choose the remaining 2 people.
Step 3: Multiply the number of ways from Step 1 and Step 2 to find the
total number of ways to choose the committee. Therefore, the total number
of ways to choose a committee of 4 people with 2 specific people included is
45 ×28 = 1260 ways.
Question 7
Question
A committee of 5 people is to be formed from a group of 12 individuals, including
5 professors and 7 students. How many different committees can be formed if
the committee must contain at least 2 professors?
Solution
Step 1: Find the number of ways to form a committee with exactly 2 professors.
There are 5ways to choose the first professor and 4ways to choose the second
professor. For the remaining 3spots, we can choose from the remaining 12−5 =
7individuals. Therefore, the number of ways to form a committee with exactly
2 professors is:
5×4×(7
3)= 5 ×4×7!
3!4! = 5 ×4×35 = 700
Step 2: Find the number of ways to form a committee with exactly 3 profes-
sors. There are (5
3)=(5
2)= 10 ways to choose 3 professors from the 5 available.
For the remaining 2 spots, we can choose from the 7 students. Therefore, the
number of ways to form a committee with exactly 3 professors is:
10 ×(7
2)= 10 ×7!
2!5! = 10 ×21 = 210
Step 3: Find the number of ways to form a committee with exactly 4 pro-
fessors. There are (5
4)= 5 ways to choose 4 professors from the 5 available. For
5
the remaining spot, we can choose from the 7 students. Therefore, the number
of ways to form a committee with exactly 4 professors is:
5×7 = 35
Step 4: Find the number of ways to form a committee with all 5 professors.
There is only (5
5)= 1 way to choose all 5 professors.
Step 5: Add up the results from Step 1, Step 2, Step 3, and Step 4 to get
the total number of ways to form a committee with at least 2 professors:
700 + 210 + 35 + 1 = 946
Question 8
Question
In a group of 10 students, how many ways can you choose a committee of 4
students if 2 of them are best friends and must be on the committee together?
Solution
Step 1: Choose the two best friends to be on the committee. There is only 1
way to choose the best friends to be on the committee together.
Step 2: Choose the remaining 2 students to fill the committee. Since we
have already chosen the two best friends, we need to choose 2 students from the
remaining 8 students. The number of ways to choose 2 students from 8 is given
by the combination formula:
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28
Step 3: Multiply the results from Step 1 and Step 2 to find the total number
of ways to choose the committee. The total number of ways to choose the
committee with the two best friends is:
1×28 = 28
Therefore, there are 28 ways to choose a committee of 4 students from a
group of 10 students where 2 of them are best friends and must be on the
committee together.
Question 9
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
In how many ways can this committee be formed if it must consist of 3 men and
2 women?
6
Solution
Step 1: Calculate the number of ways to choose 3 men out of 10. This can be
done using combinations.
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120
Step 2: Calculate the number of ways to choose 2 women out of 8. This can
also be done using combinations.
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28
Step 3: Multiply the results from Step 1 and Step 2 to find the total number
of ways to form the committee. Total number of ways = 120 ×28 = 3360
Therefore, there are 3360 ways to form a committee of 3 men and 2 women
from a group of 10 men and 8 women.
Question 10
Question
A research team is composing a committee of 5 members from a pool of 12
researchers. If 3 of the researchers are statisticians and 9 are not, how many
different committees can be formed if each committee must include at least 2
statisticians?
Solution
Step 1: Calculate the number of committees with exactly 2 statisticians: Since
there are 3 statisticians, the number of ways to choose 2 of them is (3
2)= 3. The
remaining 3 members must be chosen from the 9 non-statisticians. This can be
done in (9
3)= 84 ways. Therefore, the total number of committees with exactly
2 statisticians is 3×84 = 252.
Step 2: Calculate the number of committees with exactly 3 statisticians:
Similarly, the number of ways to choose 3 statisticians from the pool of 3 is
(3
3)= 1. The remaining 2 members must be chosen from the 9 non-statisticians.
This can be done in (9
2)= 36 ways. Therefore, the total number of committees
with exactly 3 statisticians is 1×36 = 36.
Step 3: Calculate the number of committees with 4 or 5 statisticians: Since
there are not enough statisticians left to form committees with 4 or 5 statisti-
cians, the number of such committees is 0.
Step 4: Calculate the total number of committees with at least 2 statisticians:
The total number of committees with at least 2 statisticians is the sum of the
committees with exactly 2 and exactly 3 statisticians, which is 252 + 36 = 288.
Therefore, there are 288 different committees that can be formed from a pool
of 12 researchers such that each committee must include at least 2 statisticians.
7
Question 11
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 form a committee with exactly 3
women: There are (8
3)ways to select 3 women from 8, and (10
2)ways to select
2 men from 10. Therefore, the number of ways to form the committee with
exactly 3 women is: (8
3)×(10
2)
Step 2: Calculate the number of ways to form a committee with exactly 4
women: There are (8
4)ways to select 4 women from 8, and (10
1)ways to select
1 man from 10. Therefore, the number of ways to form the committee with
exactly 4 women is: (8
4)×(10
1)
Step 3: Calculate the number of ways to form a committee with exactly 5
women: There is only (8
5)way to select 5 women from 8. Therefore, the number
of ways to form the committee with exactly 5 women is: (8
5)
Step 4: Add up the results from Step 1, Step 2, and Step 3 to get the
total number of ways to form a committee with at least 3 women: Total =
(8
3)×(10
2)+(8
4)×(10
1)+(8
5)
Calculating this expression will give us the final answer.
Question 12
Question
In a group of 10 people, how many ways can we select a committee of 4 people
where 2 must serve as co-chairs?
Solution
Step 1: First, we need to choose 2 people to serve as co-chairs from the 10
people. This can be done in (10
2)ways.
Step 2: Next, we need to choose the remaining 2 people to complete the
committee from the remaining 8 people. This can be done in (8
2)ways.
Step 3: To find the total number of ways to select the committee with 2
co-chairs, we multiply the results from Step 1 and Step 2.
Therefore, the total number of ways to select a committee of 4 people where
2 must serve as co-chairs from a group of 10 people is (10
2)·(8
2)= 45 ·28 = 1260
ways.
8
Question 13
Question
A committee of 5 people is to be formed from a group of 10 women and 8 men.
If the committee must contain at least 2 women, how many different committees
can be formed?
Solution
Step 1: Calculate the total number of ways to form a committee of 5 people
without any restrictions. This is the combination of 18 people taken 5 at a time.
Total ways to form a committee =(18
5)
Step 2: Calculate the number of ways to form a committee with no women.
This is the combination of 8 men taken 5 at a time.
Ways to form a committee with no women =(8
5)
Step 3: Calculate the number of ways to form a committee with exactly 1
woman. This involves choosing 1 woman from 10 women and 4 men from 8
men.
Ways to form a committee with 1 woman =(10
1)×(8
4)
Step 4: Calculate the total number of ways to form a committee with at
least 2 women. This is the total number of ways minus the ways with no women
and with 1 woman.
Ways to form a committee with at least 2 women =(18
5)−(8
5)−((10
1)×(8
4))
Step 5: Substitute the values and calculate the total number of ways.
Ways to form a committee with at least 2 women =(18
5)−(8
5)−((10
1)×(8
4))
Ways to form a committee with at least 2 women =18!
5!(18 −5)!−8!
5!(8 −5)!−(10!
1!(10 −1)! ×8!
4!(8 −4)!)
Ways to form a committee with at least 2 women = 856 −56 −(10 ×70)
Ways to form a committee with at least 2 women = 856 −56 −700
Ways to form a committee with at least 2 women = 100
Therefore, there are 100 different committees that can be formed with at
least 2 women.
9
Question 14
Question
In a group of 10 students, how many ways are there to select a committee of 4
students?
Solution
Step 1: To find the number of ways to select a committee of 4 students from a
group of 10, we will use the combination formula:
C(n, k) = n!
k!(n−k)!
where nis the total number of students and kis the number of students to be
selected for the committee.
Step 2: Substituting n= 10 and k= 4 into the combination formula, we
have:
C(10,4) = 10!
4!(10 −4)!
Step 3: Calculating the factorials:
C(10,4) = 10!
4!6!
C(10,4) = 10 ×9×8×7×6!
4×3×2×1×6!
Step 4: Simplifying the expression:
C(10,4) = 10 ×9×8×7
4×3×2×1
C(10,4) = 210
Step 5: Therefore, there are 210 ways to select a committee of 4 students
from a group of 10 students.
Question 15
Question
A committee of 5 people is to be formed from a group of 10 women and 8 men. If
the committee must consist of at least 3 women, how many different committees
can be formed?
10
Solution
To form a committee of 5 people with at least 3 women, we consider two cases:
1. Exactly 3 women are chosen. 2. All 5 members are women.
Case 1: Exactly 3 women are chosen.
• Selecting 3 women out of 10 can be done in (10
3)ways.
• Selecting 2 men out of 8 can be done in (8
2)ways.
• Using the Multiplication Principle, the total number of committees with
exactly 3 women is (10
3)×(8
2).
Case 2: All 5 members are women.
• Selecting 5 women out of 10 can be done in (10
5)ways.
Total number of committees with at least 3 women:
Total =(10
3)×(8
2)+(10
5)
=10!
3!7! ×8!
2!6! +10!
5!5!
= 120 ×28 + 252
= 3360 + 252
= 3612
Therefore, there are 3612 different committees that can be formed with at
least 3 women.
Question 16
Question
In a group of 10 people, how many ways can 3 people be selected to form a
committee if 2 of the people refuse to be on the committee together?
Solution
Step 1: First, we find the total number of ways to select a committee of 3 people
from a group of 10. Since order doesn’t matter, we will use the combination
formula: (n
r)=n!
r!(n−r)!
where nis the total number of people in the group and ris the number of people
chosen for the committee.
11
Plugging in n= 10 and r= 3:
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120
So, there are 120 ways to select a committee of 3 people from the group.
Step 2: Next, we find the number of ways to select a committee if 2 specific
people refuse to be on the committee together. We will subtract the number of
ways they can be together from the total.
Let’s consider the 2 people who refuse to be on the committee as one unit.
Now we have 9 individuals (8 + the ”unit”) to choose from. To select 3 people
from these 9 individuals, without choosing the ”unit”, we will use the combina-
tion formula again:
(9
3)=9!
3!(9 −3)! =9×8×7
3×2×1= 84
Step 3: Finally, we subtract the number of ways the 2 people can be together
from the total to find the number of ways to form a committee where they are
not together:
120 −84 = 36
So, there are 36 ways to select a committee of 3 people from the group of 10
where the 2 specific people refuse to be on the committee together.
Question 17
Question
In a class of 30 students, 15 are freshmen, 10 are sophomores, and 5 are juniors.
If 8 students are chosen randomly to form a group for a project, what is the
probability that the group will consist of 3 freshmen, 3 sophomores, and 2
juniors?
Solution
Step 1: Find the total number of ways to choose 8 students from a class of 30.
(30
8)=30!
8!(30 −8)! =30!
8!22! = 30,045
Step 2: Find the number of ways to choose 3 freshmen, 3 sophomores, and
2 juniors for the group.
(15
3)×(10
3)×(5
2)=15!
3!(15 −3)!×10!
3!(10 −3)!×5!
2!(5 −2)! = 455×120×10 = 546,000
12
Step 3: Find the probability that the group consists of 3 freshmen, 3 sopho-
mores, and 2 juniors.
P(3 freshmen, 3 sophomores, 2 juniors) = Number of favorable outcomes
Total number of outcomes =546,000
30,045 ≈0.1818
Therefore, the probability that the group will consist of 3 freshmen, 3 sopho-
mores, and 2 juniors is approximately 0.1818.
Question 18
Question
In a mathematics competition, a contestant has to answer 6 out of 10 multiple
choice questions. Each question has 4 possible choices. How many different
ways can the contestant answer the 6 questions if he wants to choose at least
one option from each question?
Solution
Step 1: Determine the number of ways to choose at least one option from each
of the 10 questions.
To ensure that the contestant chooses at least one option from each question,
we can break this down into cases: - Case 1: Exactly one option chosen from
each question. - Case 2: Exactly two options chosen from one question, and one
option chosen from each of the remaining 9 questions. - Case 3: Exactly two
options chosen from two different questions, and one option chosen from each
of the remaining 8 questions. - Case 4: Exactly two options chosen from three
different questions, and one option chosen from each of the remaining 7 ques-
tions. - Case 5: Exactly two options chosen from four different questions, and
one option chosen from each of the remaining 6 questions. - Case 6: One option
chosen from one question, and two options chosen from each of the remaining 5
questions.
We will calculate the number of ways for each case separately.
Step 2: Calculate the number of ways for each case.
- Case 1: 10 ×46ways - Case 2: (10
1)×(9
1)×46ways - Case 3: (10
2)×(8
1)×46
ways - Case 4: (10
3)×(7
1)×46ways - Case 5: (10
4)×(6
1)×46ways - Case 6:
(10
1)×(5
2)×46ways
Step 3: Calculate the total number of ways.
The total number of ways to answer the 6 questions with at least one option
from each question is the sum of the ways from all cases.
Total = Case 1 + Case 2 + Case 3 + Case 4 + Case 5 + Case 6.
13
Question 19
Question
In a group of 10 people, how many different ways can a committee of 3 people
be formed if one person, Alice, must be included?
Solution
Step 1: Since Alice must be included in the committee, we need to choose 2
more people from the remaining 9 people.
Step 2: The number of ways to choose 2 people from a group of 9 is given by
the combination formula (n
r)=n!
r!(n−r)! where nis the total number of people
in the group and ris the number of people to be chosen for the committee.
Step 3: Substituting n= 9 and r= 2 into the formula, we get:
(9
2)=9!
2!(9 −2)! =9!
2!7! =9×8
2×1= 36
Step 4: Therefore, there are 36 different ways to form a committee of 3
people with Alice included from a group of 10 people.
Question 20
Question
In a group of 8 people, how many ways are there to choose a team of 4 people
to represent the group in a competition?
Solution
Step 1: To find the number of ways to choose a team of 4 from a group of 8, we
will use the combination formula. The formula for combination is (n
r)=n!
r!(n−r)! ,
where nis the total number of items and ris the number of items to choose.
Step 2: Substitute n= 8 and r= 4 into the combination formula.
(8
4)=8!
4!(8 −4)!
Step 3: Calculate the factorials in the formula.
8! = 8 ×7×6×5×4×3×2×1 = 40,320
4! = 4 ×3×2×1 = 24
8−4 = 4
4! = 4 ×3×2×1 = 24
14
Step 4: Substitute the factorials back into the formula and simplify.
(8
4)=40,320
24 ×24
(8
4)=40,320
576
(8
4)= 70
Therefore, there are 70 ways to choose a team of 4 people from a group of 8
to represent the group in a competition.
Question 21
Question
In how many ways can a committee of 5 people be formed from a group of 10
men and 8 women if the committee must consist of at least 3 men?
Solution
Step 1: Calculate the number of ways to choose exactly 3, 4, or 5 men.
• Choose 3 men from 10: (10
3)=10!
3!(10−3)! = 120 ways.
• Choose 2 women from 8: (8
2)=8!
2!(8−2)! = 28 ways.
• Total ways to choose 3 men and 2 women: 120 ×28 = 3360 ways.
• Choose 4 men from 10: (10
4)=10!
4!(10−4)! = 210 ways.
• Choose 1 woman from 8: (8
1)=8!
1!(8−1)! = 8 ways.
• Total ways to choose 4 men and 1 woman: 210 ×8 = 1680 ways.
• Choose 5 men from 10: (10
5)=10!
5!(10−5)! = 252 ways.
• Choose 0 women from 8: (8
0)= 1 way.
• Total ways to choose 5 men and 0 women: 252 ×1 = 252 ways.
Step 2: Add up the total number of ways to form the committee with at
least 3 men.
Total ways with at least 3 men = 3360 + 1680 + 252
= 5292 ways
Therefore, there are 5292 ways to form a committee of 5 people from a group
of 10 men and 8 women if the committee must consist of at least 3 men.
15
Question 22
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 women, how many different committees
can be formed?
Solution
Step 1: Find the number of ways to form a committee with exactly 2 women
and 3 men. There are (8
2)ways to choose 2 women from the 8 available, and
(10
3)ways to choose 3 men from the 10 available. Therefore, the number of
committees with exactly 2 women and 3 men is (8
2)×(10
3).
Step 2: Find the number of ways to form a committee with exactly 3 women
and 2 men. Similarly, there are (8
3)ways to choose 3 women and (10
2)ways to
choose 2 men. Therefore, the number of committees with exactly 3 women and
2 men is (8
3)×(10
2).
Step 3: Find the number of ways to form a committee with all 4 or 5 women.
There are (8
4)ways to choose 4 women and (10
1)ways to choose 1 man, and
(8
5)×(10
0)ways to choose all 5 women. Therefore, the number of committees
with all 4 or 5 women is (8
4)×(10
1)+(8
5)×(10
0).
Step 4: Add the results from Step 1, Step 2, and Step 3 to find the total
number of committees. The total number of committees is the sum of commit-
tees with exactly 2 women and 3 men, exactly 3 women and 2 men, and all 4 or
5 women. Therefore, the total number of committees is (8
2)×(10
3)+(8
3)×(10
2)+
((8
4)×(10
1)+(8
5)×(10
0)). Calculate this expression to find the final answer.
Question 23
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 3 men, what is the total number of
different committees that can be formed?
Solution
Step 1: Calculate the total number of ways to form a committee with at least
3 men. Step 2: Calculate the total number of ways to form a committee with
exactly 3 men. Step 3: Calculate the total number of ways to form a committee
with exactly 4 men. Step 4: Calculate the total number of ways to form a
committee with exactly 5 men. Step 5: Add the results from Step 2, Step 3,
and Step 4 to find the total number of different committees that can be formed.
16
Step 1: To calculate the total number of ways to form a committee with at
least 3 men, we can choose 3, 4, or 5 men from the group of 10 men, and then
choose the remaining members of the committee from the group of 8 women.
Step 2: To calculate the total number of ways to form a committee with
exactly 3 men, we choose 3 men from the group of 10 men and choose the
remaining 2 members from the group of 8 women. This can be calculated using
the formula for combinations: (10
3)×(8
2)= 120 ×28 = 3360.
Step 3: To calculate the total number of ways to form a committee with
exactly 4 men, we choose 4 men from the group of 10 men and choose the
remaining member from the group of 8 women. This can be calculated using
the formula for combinations: (10
4)×(8
1)= 210 ×8 = 1680.
Step 4: To calculate the total number of ways to form a committee with
exactly 5 men, we choose all 5 men from the group of 10 men. Since there are
no women on this committee, there is only 1 way to form this committee.
Step 5: Adding the results from Step 2, Step 3, and Step 4: Total number
of different committees that can be formed = 3360 + 1680 + 1 = 5041.
Therefore, the total number of different committees that can be formed from
the group of 10 men and 8 women, with the committee consisting of at least 3
men, is 5041.
Question 24
Question
How many different ways can you arrange the letters in the word ”STATISTICS”
such that no two ’T’s are adjacent?
Solution
To solve this problem, we first find the total number of ways to arrange the
letters in ”STATISTICS”, and then subtract the number of ways where the ’T’s
are adjacent.
Step 1: Calculate the total number of ways to arrange the letters
in ”STATISTICS” The word ”STATISTICS” has 10 letters, but there are
repeating letters: 3 ’S’s, 3 ’T’s, and 2 ’I’s. Thus, the total number of ways to
arrange the letters is given by:
10!
3! ×3! ×2! = 50400
Step 2: Calculate the number of ways the ’T’s are adjacent Consider
the two ’T’s as a single entity (TT). Now we have 9 entities to arrange: TT, S,
S, S, I, I, T, I, C, S. This can be done in:
9!
2! ×3! = 30240
ways.
17
Step 3: Subtract the number of arrangements with ’T’s adjacent
from the total The number of ways to arrange the letters in ”STATISTICS”
such that no two ’T’s are adjacent is:
50400 −30240 = 20160
Therefore, there are 20,160 different ways to arrange the letters in ”STATIS-
TICS” such that no two ’T’s are adjacent.
Question 25
Question
In a group of 10 students, how many ways can we choose 3 students to represent
the group in a competition?
Solution
Step 1: We can solve this problem using the combination formula. The number
of ways to choose kout of nobjects, denoted as C(n, k)or (n
k), is given by:
C(n, k) = n!
k!(n−k)!
Step 2: In this case, we have 10 students and we want to choose 3 to represent
the group. Using the combination formula, we have:
(10
3)=10!
3!(10 −3)!
Step 3: Calculating the factorials, we get:
(10
3)=10 ×9×8
3×2×1
Step 4: Simplifying, we find:
(10
3)= 120
Therefore, there are 120 ways to choose 3 students out of 10 to represent the
group in a competition.
18
Step 4: Calculate the total number of ways to form a committee with 3 men
and 2 women. The total number of ways is given by the product of the number
of ways to choose 3 men and 2 women:
1×21 = 21
Step 5: Determine the probability of selecting a committee with 3 men and
2 women. The probability is the ratio of the number of favorable outcomes to
the total number of outcomes:
P(3 men, 2 women) = 21
252 =1
12
Therefore, the probability that the committee consists of exactly 3 men and
2 women is 1
12 .
Question 2
Question
A committee of 5 people is to be formed from a group of 8 men and 5 women.
If at least 2 men must be on the committee, how many different ways can the
committee be formed?
Solution
Step 1: Find the total number of ways to form a committee with 5 people.
Given that there are 8 men and 5 women, the total number of ways to form a
committee of 5 people from this group is given by the combination formula:
(8+5
5)=(13
5)=13!
5!(13 −5)! = 1287
Step 2: Find the number of ways to form a committee with no men. Since
at least 2 men must be on the committee, find the number of ways to form a
committee with no men first. This will be a committee of 5 women, which is
given by the combination formula:
(5
5)= 1
Step 3: Find the number of ways to form a committee with 1 man. Similarly,
find the number of ways to form a committee with 1 man and 4 women, using
the combination formula:
(8
1)×(5
4)= 8 ×5 = 40
Step 4: Calculate the total number of ways to form a committee with at
least 2 men. To find the number of ways to form a committee with at least 2
2
men, subtract the number of ways with no men and the number of ways with 1
man from the total number of ways:
1287 −1−40 = 1246
Therefore, there are 1246 different ways to form the committee with at least
2 men.
Question 3
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
How many different committees can be formed if the committee must consist of
at least 3 women?
Solution
Step 1: First, we’ll consider the number of ways to choose a committee with
exactly 3 women. Step 2: Select 3 women from the group of 8 women: (8
3)= 56
ways. Step 3: Select the remaining 2 people from the group of 10 men: (10
2)= 45
ways. Step 4: The total number of ways to form a committee with exactly 3
women is 56 ×45 = 2520 ways.
Step 5: Next, we’ll consider the number of ways to choose a committee with
exactly 4 women. Step 6: Select 4 women from the group of 8 women: (8
4)= 70
ways. Step 7: Select the remaining 1 person from the group of 10 men: (10
1)= 10
ways. Step 8: The total number of ways to form a committee with exactly 4
women is 70 ×10 = 700 ways.
Step 9: Finally, we’ll consider the number of ways to choose a committee
with all 5 women. Step 10: Select all 5 women from the group of 8 women:
(8
5)= 56 ways. Step 11: Since there are no men in this case, the remaining 0
people are automatically assigned. Step 12: The total number of ways to form
a committee with all 5 women is 56 ×1 = 56 ways.
Step 13: Therefore, the total number of different committees that can be
formed if the committee must consist of at least 3 women is 2520 + 700 + 56 =
3276 ways.
Question 4
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
How many ways can the committee be formed if it must include at least 3
women?
3
Solution
Step 1: Calculate the number of ways to choose 3 women from the 8 available.
There are (8
3)=8!
3!(8−3)! = 56 ways to choose 3 women from the group.
Step 2: Calculate the number of ways to choose the remaining 2 people from
the remaining pool of 10 men and 5 women (after selecting 3 women). There
are (10+5
2)=(15
2)=15!
2!(15−2)! = 105 ways to choose 2 people from this group.
Step 3: Calculate the total number of ways to form the committee with at
least 3 women. Multiply the results from Step 1 and Step 2 to get the total
number of ways to form the committee: 56 ×105 = 5880 ways.
Therefore, there are 5880 ways to form the committee with at least 3 women
from the group of 10 men and 8 women.
Question 5
Question
A committee of 5 people is to be formed from a group of 8 men and 6 women. If
the committee must consist of at least 2 men and 2 women, how many different
committees can be formed?
Solution
Step 1: Calculate the number of ways to choose 2 men out of 8 men. There are
(8
2)ways to choose 2 men from 8 men. This can be calculated as:
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28
Step 2: Calculate the number of ways to choose 2 women out of 6 women.
There are (6
2)ways to choose 2 women from 6 women. This can be calculated
as: (6
2)=6!
2!(6 −2)! =6×5
2×1= 15
Step 3: Calculate the number of ways to choose the remaining person (either
man or woman). There are 4 remaining people to choose from (6 men and 4
women). Therefore, there are 4 ways to choose the remaining person.
Step 4: Calculate the total number of committees that can be formed. The
total number of committees that can be formed is the product of the number
of ways to choose 2 men, 2 women, and the remaining person. Hence, the total
number of committees is:
28 ×15 ×4 = 1680
Therefore, there are 1680 different committees that can be formed with at
least 2 men and 2 women from the given group of people.
4
Question 6
Question
In a group of 10 people, how many ways can we choose a committee of 4 people
if 2 specific people must be included in the committee?
Solution
Step 1: Choose the 2 specific people who must be included in the committee.
There are (10
2)=10!
2!(10−2)! = 45 ways to choose the 2 specific people.
Step 2: Choose the remaining 2 people to complete the committee. After
selecting the 2 specific people, there are 8 people left to choose from. So, there
are (8
2)=8!
2!(8−2)! = 28 ways to choose the remaining 2 people.
Step 3: Multiply the number of ways from Step 1 and Step 2 to find the
total number of ways to choose the committee. Therefore, the total number
of ways to choose a committee of 4 people with 2 specific people included is
45 ×28 = 1260 ways.
Question 7
Question
A committee of 5 people is to be formed from a group of 12 individuals, including
5 professors and 7 students. How many different committees can be formed if
the committee must contain at least 2 professors?
Solution
Step 1: Find the number of ways to form a committee with exactly 2 professors.
There are 5ways to choose the first professor and 4ways to choose the second
professor. For the remaining 3spots, we can choose from the remaining 12−5 =
7individuals. Therefore, the number of ways to form a committee with exactly
2 professors is:
5×4×(7
3)= 5 ×4×7!
3!4! = 5 ×4×35 = 700
Step 2: Find the number of ways to form a committee with exactly 3 profes-
sors. There are (5
3)=(5
2)= 10 ways to choose 3 professors from the 5 available.
For the remaining 2 spots, we can choose from the 7 students. Therefore, the
number of ways to form a committee with exactly 3 professors is:
10 ×(7
2)= 10 ×7!
2!5! = 10 ×21 = 210
Step 3: Find the number of ways to form a committee with exactly 4 pro-
fessors. There are (5
4)= 5 ways to choose 4 professors from the 5 available. For
5
the remaining spot, we can choose from the 7 students. Therefore, the number
of ways to form a committee with exactly 4 professors is:
5×7 = 35
Step 4: Find the number of ways to form a committee with all 5 professors.
There is only (5
5)= 1 way to choose all 5 professors.
Step 5: Add up the results from Step 1, Step 2, Step 3, and Step 4 to get
the total number of ways to form a committee with at least 2 professors:
700 + 210 + 35 + 1 = 946
Question 8
Question
In a group of 10 students, how many ways can you choose a committee of 4
students if 2 of them are best friends and must be on the committee together?
Solution
Step 1: Choose the two best friends to be on the committee. There is only 1
way to choose the best friends to be on the committee together.
Step 2: Choose the remaining 2 students to fill the committee. Since we
have already chosen the two best friends, we need to choose 2 students from the
remaining 8 students. The number of ways to choose 2 students from 8 is given
by the combination formula:
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28
Step 3: Multiply the results from Step 1 and Step 2 to find the total number
of ways to choose the committee. The total number of ways to choose the
committee with the two best friends is:
1×28 = 28
Therefore, there are 28 ways to choose a committee of 4 students from a
group of 10 students where 2 of them are best friends and must be on the
committee together.
Question 9
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
In how many ways can this committee be formed if it must consist of 3 men and
2 women?
6
Solution
Step 1: Calculate the number of ways to choose 3 men out of 10. This can be
done using combinations.
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120
Step 2: Calculate the number of ways to choose 2 women out of 8. This can
also be done using combinations.
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28
Step 3: Multiply the results from Step 1 and Step 2 to find the total number
of ways to form the committee. Total number of ways = 120 ×28 = 3360
Therefore, there are 3360 ways to form a committee of 3 men and 2 women
from a group of 10 men and 8 women.
Question 10
Question
A research team is composing a committee of 5 members from a pool of 12
researchers. If 3 of the researchers are statisticians and 9 are not, how many
different committees can be formed if each committee must include at least 2
statisticians?
Solution
Step 1: Calculate the number of committees with exactly 2 statisticians: Since
there are 3 statisticians, the number of ways to choose 2 of them is (3
2)= 3. The
remaining 3 members must be chosen from the 9 non-statisticians. This can be
done in (9
3)= 84 ways. Therefore, the total number of committees with exactly
2 statisticians is 3×84 = 252.
Step 2: Calculate the number of committees with exactly 3 statisticians:
Similarly, the number of ways to choose 3 statisticians from the pool of 3 is
(3
3)= 1. The remaining 2 members must be chosen from the 9 non-statisticians.
This can be done in (9
2)= 36 ways. Therefore, the total number of committees
with exactly 3 statisticians is 1×36 = 36.
Step 3: Calculate the number of committees with 4 or 5 statisticians: Since
there are not enough statisticians left to form committees with 4 or 5 statisti-
cians, the number of such committees is 0.
Step 4: Calculate the total number of committees with at least 2 statisticians:
The total number of committees with at least 2 statisticians is the sum of the
committees with exactly 2 and exactly 3 statisticians, which is 252 + 36 = 288.
Therefore, there are 288 different committees that can be formed from a pool
of 12 researchers such that each committee must include at least 2 statisticians.
7
Question 11
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 form a committee with exactly 3
women: There are (8
3)ways to select 3 women from 8, and (10
2)ways to select
2 men from 10. Therefore, the number of ways to form the committee with
exactly 3 women is: (8
3)×(10
2)
Step 2: Calculate the number of ways to form a committee with exactly 4
women: There are (8
4)ways to select 4 women from 8, and (10
1)ways to select
1 man from 10. Therefore, the number of ways to form the committee with
exactly 4 women is: (8
4)×(10
1)
Step 3: Calculate the number of ways to form a committee with exactly 5
women: There is only (8
5)way to select 5 women from 8. Therefore, the number
of ways to form the committee with exactly 5 women is: (8
5)
Step 4: Add up the results from Step 1, Step 2, and Step 3 to get the
total number of ways to form a committee with at least 3 women: Total =
(8
3)×(10
2)+(8
4)×(10
1)+(8
5)
Calculating this expression will give us the final answer.
Question 12
Question
In a group of 10 people, how many ways can we select a committee of 4 people
where 2 must serve as co-chairs?
Solution
Step 1: First, we need to choose 2 people to serve as co-chairs from the 10
people. This can be done in (10
2)ways.
Step 2: Next, we need to choose the remaining 2 people to complete the
committee from the remaining 8 people. This can be done in (8
2)ways.
Step 3: To find the total number of ways to select the committee with 2
co-chairs, we multiply the results from Step 1 and Step 2.
Therefore, the total number of ways to select a committee of 4 people where
2 must serve as co-chairs from a group of 10 people is (10
2)·(8
2)= 45 ·28 = 1260
ways.
8
Question 13
Question
A committee of 5 people is to be formed from a group of 10 women and 8 men.
If the committee must contain at least 2 women, how many different committees
can be formed?
Solution
Step 1: Calculate the total number of ways to form a committee of 5 people
without any restrictions. This is the combination of 18 people taken 5 at a time.
Total ways to form a committee =(18
5)
Step 2: Calculate the number of ways to form a committee with no women.
This is the combination of 8 men taken 5 at a time.
Ways to form a committee with no women =(8
5)
Step 3: Calculate the number of ways to form a committee with exactly 1
woman. This involves choosing 1 woman from 10 women and 4 men from 8
men.
Ways to form a committee with 1 woman =(10
1)×(8
4)
Step 4: Calculate the total number of ways to form a committee with at
least 2 women. This is the total number of ways minus the ways with no women
and with 1 woman.
Ways to form a committee with at least 2 women =(18
5)−(8
5)−((10
1)×(8
4))
Step 5: Substitute the values and calculate the total number of ways.
Ways to form a committee with at least 2 women =(18
5)−(8
5)−((10
1)×(8
4))
Ways to form a committee with at least 2 women =18!
5!(18 −5)!−8!
5!(8 −5)!−(10!
1!(10 −1)! ×8!
4!(8 −4)!)
Ways to form a committee with at least 2 women = 856 −56 −(10 ×70)
Ways to form a committee with at least 2 women = 856 −56 −700
Ways to form a committee with at least 2 women = 100
Therefore, there are 100 different committees that can be formed with at
least 2 women.
9
Question 14
Question
In a group of 10 students, how many ways are there to select a committee of 4
students?
Solution
Step 1: To find the number of ways to select a committee of 4 students from a
group of 10, we will use the combination formula:
C(n, k) = n!
k!(n−k)!
where nis the total number of students and kis the number of students to be
selected for the committee.
Step 2: Substituting n= 10 and k= 4 into the combination formula, we
have:
C(10,4) = 10!
4!(10 −4)!
Step 3: Calculating the factorials:
C(10,4) = 10!
4!6!
C(10,4) = 10 ×9×8×7×6!
4×3×2×1×6!
Step 4: Simplifying the expression:
C(10,4) = 10 ×9×8×7
4×3×2×1
C(10,4) = 210
Step 5: Therefore, there are 210 ways to select a committee of 4 students
from a group of 10 students.
Question 15
Question
A committee of 5 people is to be formed from a group of 10 women and 8 men. If
the committee must consist of at least 3 women, how many different committees
can be formed?
10
Solution
To form a committee of 5 people with at least 3 women, we consider two cases:
1. Exactly 3 women are chosen. 2. All 5 members are women.
Case 1: Exactly 3 women are chosen.
• Selecting 3 women out of 10 can be done in (10
3)ways.
• Selecting 2 men out of 8 can be done in (8
2)ways.
• Using the Multiplication Principle, the total number of committees with
exactly 3 women is (10
3)×(8
2).
Case 2: All 5 members are women.
• Selecting 5 women out of 10 can be done in (10
5)ways.
Total number of committees with at least 3 women:
Total =(10
3)×(8
2)+(10
5)
=10!
3!7! ×8!
2!6! +10!
5!5!
= 120 ×28 + 252
= 3360 + 252
= 3612
Therefore, there are 3612 different committees that can be formed with at
least 3 women.
Question 16
Question
In a group of 10 people, how many ways can 3 people be selected to form a
committee if 2 of the people refuse to be on the committee together?
Solution
Step 1: First, we find the total number of ways to select a committee of 3 people
from a group of 10. Since order doesn’t matter, we will use the combination
formula: (n
r)=n!
r!(n−r)!
where nis the total number of people in the group and ris the number of people
chosen for the committee.
11
Plugging in n= 10 and r= 3:
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120
So, there are 120 ways to select a committee of 3 people from the group.
Step 2: Next, we find the number of ways to select a committee if 2 specific
people refuse to be on the committee together. We will subtract the number of
ways they can be together from the total.
Let’s consider the 2 people who refuse to be on the committee as one unit.
Now we have 9 individuals (8 + the ”unit”) to choose from. To select 3 people
from these 9 individuals, without choosing the ”unit”, we will use the combina-
tion formula again:
(9
3)=9!
3!(9 −3)! =9×8×7
3×2×1= 84
Step 3: Finally, we subtract the number of ways the 2 people can be together
from the total to find the number of ways to form a committee where they are
not together:
120 −84 = 36
So, there are 36 ways to select a committee of 3 people from the group of 10
where the 2 specific people refuse to be on the committee together.
Question 17
Question
In a class of 30 students, 15 are freshmen, 10 are sophomores, and 5 are juniors.
If 8 students are chosen randomly to form a group for a project, what is the
probability that the group will consist of 3 freshmen, 3 sophomores, and 2
juniors?
Solution
Step 1: Find the total number of ways to choose 8 students from a class of 30.
(30
8)=30!
8!(30 −8)! =30!
8!22! = 30,045
Step 2: Find the number of ways to choose 3 freshmen, 3 sophomores, and
2 juniors for the group.
(15
3)×(10
3)×(5
2)=15!
3!(15 −3)!×10!
3!(10 −3)!×5!
2!(5 −2)! = 455×120×10 = 546,000
12
Step 3: Find the probability that the group consists of 3 freshmen, 3 sopho-
mores, and 2 juniors.
P(3 freshmen, 3 sophomores, 2 juniors) = Number of favorable outcomes
Total number of outcomes =546,000
30,045 ≈0.1818
Therefore, the probability that the group will consist of 3 freshmen, 3 sopho-
mores, and 2 juniors is approximately 0.1818.
Question 18
Question
In a mathematics competition, a contestant has to answer 6 out of 10 multiple
choice questions. Each question has 4 possible choices. How many different
ways can the contestant answer the 6 questions if he wants to choose at least
one option from each question?
Solution
Step 1: Determine the number of ways to choose at least one option from each
of the 10 questions.
To ensure that the contestant chooses at least one option from each question,
we can break this down into cases: - Case 1: Exactly one option chosen from
each question. - Case 2: Exactly two options chosen from one question, and one
option chosen from each of the remaining 9 questions. - Case 3: Exactly two
options chosen from two different questions, and one option chosen from each
of the remaining 8 questions. - Case 4: Exactly two options chosen from three
different questions, and one option chosen from each of the remaining 7 ques-
tions. - Case 5: Exactly two options chosen from four different questions, and
one option chosen from each of the remaining 6 questions. - Case 6: One option
chosen from one question, and two options chosen from each of the remaining 5
questions.
We will calculate the number of ways for each case separately.
Step 2: Calculate the number of ways for each case.
- Case 1: 10 ×46ways - Case 2: (10
1)×(9
1)×46ways - Case 3: (10
2)×(8
1)×46
ways - Case 4: (10
3)×(7
1)×46ways - Case 5: (10
4)×(6
1)×46ways - Case 6:
(10
1)×(5
2)×46ways
Step 3: Calculate the total number of ways.
The total number of ways to answer the 6 questions with at least one option
from each question is the sum of the ways from all cases.
Total = Case 1 + Case 2 + Case 3 + Case 4 + Case 5 + Case 6.
13
Question 19
Question
In a group of 10 people, how many different ways can a committee of 3 people
be formed if one person, Alice, must be included?
Solution
Step 1: Since Alice must be included in the committee, we need to choose 2
more people from the remaining 9 people.
Step 2: The number of ways to choose 2 people from a group of 9 is given by
the combination formula (n
r)=n!
r!(n−r)! where nis the total number of people
in the group and ris the number of people to be chosen for the committee.
Step 3: Substituting n= 9 and r= 2 into the formula, we get:
(9
2)=9!
2!(9 −2)! =9!
2!7! =9×8
2×1= 36
Step 4: Therefore, there are 36 different ways to form a committee of 3
people with Alice included from a group of 10 people.
Question 20
Question
In a group of 8 people, how many ways are there to choose a team of 4 people
to represent the group in a competition?
Solution
Step 1: To find the number of ways to choose a team of 4 from a group of 8, we
will use the combination formula. The formula for combination is (n
r)=n!
r!(n−r)! ,
where nis the total number of items and ris the number of items to choose.
Step 2: Substitute n= 8 and r= 4 into the combination formula.
(8
4)=8!
4!(8 −4)!
Step 3: Calculate the factorials in the formula.
8! = 8 ×7×6×5×4×3×2×1 = 40,320
4! = 4 ×3×2×1 = 24
8−4 = 4
4! = 4 ×3×2×1 = 24
14
Step 4: Substitute the factorials back into the formula and simplify.
(8
4)=40,320
24 ×24
(8
4)=40,320
576
(8
4)= 70
Therefore, there are 70 ways to choose a team of 4 people from a group of 8
to represent the group in a competition.
Question 21
Question
In how many ways can a committee of 5 people be formed from a group of 10
men and 8 women if the committee must consist of at least 3 men?
Solution
Step 1: Calculate the number of ways to choose exactly 3, 4, or 5 men.
• Choose 3 men from 10: (10
3)=10!
3!(10−3)! = 120 ways.
• Choose 2 women from 8: (8
2)=8!
2!(8−2)! = 28 ways.
• Total ways to choose 3 men and 2 women: 120 ×28 = 3360 ways.
• Choose 4 men from 10: (10
4)=10!
4!(10−4)! = 210 ways.
• Choose 1 woman from 8: (8
1)=8!
1!(8−1)! = 8 ways.
• Total ways to choose 4 men and 1 woman: 210 ×8 = 1680 ways.
• Choose 5 men from 10: (10
5)=10!
5!(10−5)! = 252 ways.
• Choose 0 women from 8: (8
0)= 1 way.
• Total ways to choose 5 men and 0 women: 252 ×1 = 252 ways.
Step 2: Add up the total number of ways to form the committee with at
least 3 men.
Total ways with at least 3 men = 3360 + 1680 + 252
= 5292 ways
Therefore, there are 5292 ways to form a committee of 5 people from a group
of 10 men and 8 women if the committee must consist of at least 3 men.
15
Question 22
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 women, how many different committees
can be formed?
Solution
Step 1: Find the number of ways to form a committee with exactly 2 women
and 3 men. There are (8
2)ways to choose 2 women from the 8 available, and
(10
3)ways to choose 3 men from the 10 available. Therefore, the number of
committees with exactly 2 women and 3 men is (8
2)×(10
3).
Step 2: Find the number of ways to form a committee with exactly 3 women
and 2 men. Similarly, there are (8
3)ways to choose 3 women and (10
2)ways to
choose 2 men. Therefore, the number of committees with exactly 3 women and
2 men is (8
3)×(10
2).
Step 3: Find the number of ways to form a committee with all 4 or 5 women.
There are (8
4)ways to choose 4 women and (10
1)ways to choose 1 man, and
(8
5)×(10
0)ways to choose all 5 women. Therefore, the number of committees
with all 4 or 5 women is (8
4)×(10
1)+(8
5)×(10
0).
Step 4: Add the results from Step 1, Step 2, and Step 3 to find the total
number of committees. The total number of committees is the sum of commit-
tees with exactly 2 women and 3 men, exactly 3 women and 2 men, and all 4 or
5 women. Therefore, the total number of committees is (8
2)×(10
3)+(8
3)×(10
2)+
((8
4)×(10
1)+(8
5)×(10
0)). Calculate this expression to find the final answer.
Question 23
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 3 men, what is the total number of
different committees that can be formed?
Solution
Step 1: Calculate the total number of ways to form a committee with at least
3 men. Step 2: Calculate the total number of ways to form a committee with
exactly 3 men. Step 3: Calculate the total number of ways to form a committee
with exactly 4 men. Step 4: Calculate the total number of ways to form a
committee with exactly 5 men. Step 5: Add the results from Step 2, Step 3,
and Step 4 to find the total number of different committees that can be formed.
16
Step 1: To calculate the total number of ways to form a committee with at
least 3 men, we can choose 3, 4, or 5 men from the group of 10 men, and then
choose the remaining members of the committee from the group of 8 women.
Step 2: To calculate the total number of ways to form a committee with
exactly 3 men, we choose 3 men from the group of 10 men and choose the
remaining 2 members from the group of 8 women. This can be calculated using
the formula for combinations: (10
3)×(8
2)= 120 ×28 = 3360.
Step 3: To calculate the total number of ways to form a committee with
exactly 4 men, we choose 4 men from the group of 10 men and choose the
remaining member from the group of 8 women. This can be calculated using
the formula for combinations: (10
4)×(8
1)= 210 ×8 = 1680.
Step 4: To calculate the total number of ways to form a committee with
exactly 5 men, we choose all 5 men from the group of 10 men. Since there are
no women on this committee, there is only 1 way to form this committee.
Step 5: Adding the results from Step 2, Step 3, and Step 4: Total number
of different committees that can be formed = 3360 + 1680 + 1 = 5041.
Therefore, the total number of different committees that can be formed from
the group of 10 men and 8 women, with the committee consisting of at least 3
men, is 5041.
Question 24
Question
How many different ways can you arrange the letters in the word ”STATISTICS”
such that no two ’T’s are adjacent?
Solution
To solve this problem, we first find the total number of ways to arrange the
letters in ”STATISTICS”, and then subtract the number of ways where the ’T’s
are adjacent.
Step 1: Calculate the total number of ways to arrange the letters
in ”STATISTICS” The word ”STATISTICS” has 10 letters, but there are
repeating letters: 3 ’S’s, 3 ’T’s, and 2 ’I’s. Thus, the total number of ways to
arrange the letters is given by:
10!
3! ×3! ×2! = 50400
Step 2: Calculate the number of ways the ’T’s are adjacent Consider
the two ’T’s as a single entity (TT). Now we have 9 entities to arrange: TT, S,
S, S, I, I, T, I, C, S. This can be done in:
9!
2! ×3! = 30240
ways.
17
Step 3: Subtract the number of arrangements with ’T’s adjacent
from the total The number of ways to arrange the letters in ”STATISTICS”
such that no two ’T’s are adjacent is:
50400 −30240 = 20160
Therefore, there are 20,160 different ways to arrange the letters in ”STATIS-
TICS” such that no two ’T’s are adjacent.
Question 25
Question
In a group of 10 students, how many ways can we choose 3 students to represent
the group in a competition?
Solution
Step 1: We can solve this problem using the combination formula. The number
of ways to choose kout of nobjects, denoted as C(n, k)or (n
k), is given by:
C(n, k) = n!
k!(n−k)!
Step 2: In this case, we have 10 students and we want to choose 3 to represent
the group. Using the combination formula, we have:
(10
3)=10!
3!(10 −3)!
Step 3: Calculating the factorials, we get:
(10
3)=10 ×9×8
3×2×1
Step 4: Simplifying, we find:
(10
3)= 120
Therefore, there are 120 ways to choose 3 students out of 10 to represent the
group in a competition.
18
Step 4: Calculate the total number of ways to form a committee with 3 men
and 2 women. The total number of ways is given by the product of the number
of ways to choose 3 men and 2 women:
1×21 = 21
Step 5: Determine the probability of selecting a committee with 3 men and
2 women. The probability is the ratio of the number of favorable outcomes to
the total number of outcomes:
P(3 men, 2 women) = 21
252 =1
12
Therefore, the probability that the committee consists of exactly 3 men and
2 women is 1
12 .
Question 2
Question
A committee of 5 people is to be formed from a group of 8 men and 5 women.
If at least 2 men must be on the committee, how many different ways can the
committee be formed?
Solution
Step 1: Find the total number of ways to form a committee with 5 people.
Given that there are 8 men and 5 women, the total number of ways to form a
committee of 5 people from this group is given by the combination formula:
(8+5
5)=(13
5)=13!
5!(13 −5)! = 1287
Step 2: Find the number of ways to form a committee with no men. Since
at least 2 men must be on the committee, find the number of ways to form a
committee with no men first. This will be a committee of 5 women, which is
given by the combination formula:
(5
5)= 1
Step 3: Find the number of ways to form a committee with 1 man. Similarly,
find the number of ways to form a committee with 1 man and 4 women, using
the combination formula:
(8
1)×(5
4)= 8 ×5 = 40
Step 4: Calculate the total number of ways to form a committee with at
least 2 men. To find the number of ways to form a committee with at least 2
2
men, subtract the number of ways with no men and the number of ways with 1
man from the total number of ways:
1287 −1−40 = 1246
Therefore, there are 1246 different ways to form the committee with at least
2 men.
Question 3
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
How many different committees can be formed if the committee must consist of
at least 3 women?
Solution
Step 1: First, we’ll consider the number of ways to choose a committee with
exactly 3 women. Step 2: Select 3 women from the group of 8 women: (8
3)= 56
ways. Step 3: Select the remaining 2 people from the group of 10 men: (10
2)= 45
ways. Step 4: The total number of ways to form a committee with exactly 3
women is 56 ×45 = 2520 ways.
Step 5: Next, we’ll consider the number of ways to choose a committee with
exactly 4 women. Step 6: Select 4 women from the group of 8 women: (8
4)= 70
ways. Step 7: Select the remaining 1 person from the group of 10 men: (10
1)= 10
ways. Step 8: The total number of ways to form a committee with exactly 4
women is 70 ×10 = 700 ways.
Step 9: Finally, we’ll consider the number of ways to choose a committee
with all 5 women. Step 10: Select all 5 women from the group of 8 women:
(8
5)= 56 ways. Step 11: Since there are no men in this case, the remaining 0
people are automatically assigned. Step 12: The total number of ways to form
a committee with all 5 women is 56 ×1 = 56 ways.
Step 13: Therefore, the total number of different committees that can be
formed if the committee must consist of at least 3 women is 2520 + 700 + 56 =
3276 ways.
Question 4
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
How many ways can the committee be formed if it must include at least 3
women?
3
Solution
Step 1: Calculate the number of ways to choose 3 women from the 8 available.
There are (8
3)=8!
3!(8−3)! = 56 ways to choose 3 women from the group.
Step 2: Calculate the number of ways to choose the remaining 2 people from
the remaining pool of 10 men and 5 women (after selecting 3 women). There
are (10+5
2)=(15
2)=15!
2!(15−2)! = 105 ways to choose 2 people from this group.
Step 3: Calculate the total number of ways to form the committee with at
least 3 women. Multiply the results from Step 1 and Step 2 to get the total
number of ways to form the committee: 56 ×105 = 5880 ways.
Therefore, there are 5880 ways to form the committee with at least 3 women
from the group of 10 men and 8 women.
Question 5
Question
A committee of 5 people is to be formed from a group of 8 men and 6 women. If
the committee must consist of at least 2 men and 2 women, how many different
committees can be formed?
Solution
Step 1: Calculate the number of ways to choose 2 men out of 8 men. There are
(8
2)ways to choose 2 men from 8 men. This can be calculated as:
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28
Step 2: Calculate the number of ways to choose 2 women out of 6 women.
There are (6
2)ways to choose 2 women from 6 women. This can be calculated
as: (6
2)=6!
2!(6 −2)! =6×5
2×1= 15
Step 3: Calculate the number of ways to choose the remaining person (either
man or woman). There are 4 remaining people to choose from (6 men and 4
women). Therefore, there are 4 ways to choose the remaining person.
Step 4: Calculate the total number of committees that can be formed. The
total number of committees that can be formed is the product of the number
of ways to choose 2 men, 2 women, and the remaining person. Hence, the total
number of committees is:
28 ×15 ×4 = 1680
Therefore, there are 1680 different committees that can be formed with at
least 2 men and 2 women from the given group of people.
4
Question 6
Question
In a group of 10 people, how many ways can we choose a committee of 4 people
if 2 specific people must be included in the committee?
Solution
Step 1: Choose the 2 specific people who must be included in the committee.
There are (10
2)=10!
2!(10−2)! = 45 ways to choose the 2 specific people.
Step 2: Choose the remaining 2 people to complete the committee. After
selecting the 2 specific people, there are 8 people left to choose from. So, there
are (8
2)=8!
2!(8−2)! = 28 ways to choose the remaining 2 people.
Step 3: Multiply the number of ways from Step 1 and Step 2 to find the
total number of ways to choose the committee. Therefore, the total number
of ways to choose a committee of 4 people with 2 specific people included is
45 ×28 = 1260 ways.
Question 7
Question
A committee of 5 people is to be formed from a group of 12 individuals, including
5 professors and 7 students. How many different committees can be formed if
the committee must contain at least 2 professors?
Solution
Step 1: Find the number of ways to form a committee with exactly 2 professors.
There are 5ways to choose the first professor and 4ways to choose the second
professor. For the remaining 3spots, we can choose from the remaining 12−5 =
7individuals. Therefore, the number of ways to form a committee with exactly
2 professors is:
5×4×(7
3)= 5 ×4×7!
3!4! = 5 ×4×35 = 700
Step 2: Find the number of ways to form a committee with exactly 3 profes-
sors. There are (5
3)=(5
2)= 10 ways to choose 3 professors from the 5 available.
For the remaining 2 spots, we can choose from the 7 students. Therefore, the
number of ways to form a committee with exactly 3 professors is:
10 ×(7
2)= 10 ×7!
2!5! = 10 ×21 = 210
Step 3: Find the number of ways to form a committee with exactly 4 pro-
fessors. There are (5
4)= 5 ways to choose 4 professors from the 5 available. For
5
the remaining spot, we can choose from the 7 students. Therefore, the number
of ways to form a committee with exactly 4 professors is:
5×7 = 35
Step 4: Find the number of ways to form a committee with all 5 professors.
There is only (5
5)= 1 way to choose all 5 professors.
Step 5: Add up the results from Step 1, Step 2, Step 3, and Step 4 to get
the total number of ways to form a committee with at least 2 professors:
700 + 210 + 35 + 1 = 946
Question 8
Question
In a group of 10 students, how many ways can you choose a committee of 4
students if 2 of them are best friends and must be on the committee together?
Solution
Step 1: Choose the two best friends to be on the committee. There is only 1
way to choose the best friends to be on the committee together.
Step 2: Choose the remaining 2 students to fill the committee. Since we
have already chosen the two best friends, we need to choose 2 students from the
remaining 8 students. The number of ways to choose 2 students from 8 is given
by the combination formula:
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28
Step 3: Multiply the results from Step 1 and Step 2 to find the total number
of ways to choose the committee. The total number of ways to choose the
committee with the two best friends is:
1×28 = 28
Therefore, there are 28 ways to choose a committee of 4 students from a
group of 10 students where 2 of them are best friends and must be on the
committee together.
Question 9
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
In how many ways can this committee be formed if it must consist of 3 men and
2 women?
6
Solution
Step 1: Calculate the number of ways to choose 3 men out of 10. This can be
done using combinations.
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120
Step 2: Calculate the number of ways to choose 2 women out of 8. This can
also be done using combinations.
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28
Step 3: Multiply the results from Step 1 and Step 2 to find the total number
of ways to form the committee. Total number of ways = 120 ×28 = 3360
Therefore, there are 3360 ways to form a committee of 3 men and 2 women
from a group of 10 men and 8 women.
Question 10
Question
A research team is composing a committee of 5 members from a pool of 12
researchers. If 3 of the researchers are statisticians and 9 are not, how many
different committees can be formed if each committee must include at least 2
statisticians?
Solution
Step 1: Calculate the number of committees with exactly 2 statisticians: Since
there are 3 statisticians, the number of ways to choose 2 of them is (3
2)= 3. The
remaining 3 members must be chosen from the 9 non-statisticians. This can be
done in (9
3)= 84 ways. Therefore, the total number of committees with exactly
2 statisticians is 3×84 = 252.
Step 2: Calculate the number of committees with exactly 3 statisticians:
Similarly, the number of ways to choose 3 statisticians from the pool of 3 is
(3
3)= 1. The remaining 2 members must be chosen from the 9 non-statisticians.
This can be done in (9
2)= 36 ways. Therefore, the total number of committees
with exactly 3 statisticians is 1×36 = 36.
Step 3: Calculate the number of committees with 4 or 5 statisticians: Since
there are not enough statisticians left to form committees with 4 or 5 statisti-
cians, the number of such committees is 0.
Step 4: Calculate the total number of committees with at least 2 statisticians:
The total number of committees with at least 2 statisticians is the sum of the
committees with exactly 2 and exactly 3 statisticians, which is 252 + 36 = 288.
Therefore, there are 288 different committees that can be formed from a pool
of 12 researchers such that each committee must include at least 2 statisticians.
7
Question 11
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 form a committee with exactly 3
women: There are (8
3)ways to select 3 women from 8, and (10
2)ways to select
2 men from 10. Therefore, the number of ways to form the committee with
exactly 3 women is: (8
3)×(10
2)
Step 2: Calculate the number of ways to form a committee with exactly 4
women: There are (8
4)ways to select 4 women from 8, and (10
1)ways to select
1 man from 10. Therefore, the number of ways to form the committee with
exactly 4 women is: (8
4)×(10
1)
Step 3: Calculate the number of ways to form a committee with exactly 5
women: There is only (8
5)way to select 5 women from 8. Therefore, the number
of ways to form the committee with exactly 5 women is: (8
5)
Step 4: Add up the results from Step 1, Step 2, and Step 3 to get the
total number of ways to form a committee with at least 3 women: Total =
(8
3)×(10
2)+(8
4)×(10
1)+(8
5)
Calculating this expression will give us the final answer.
Question 12
Question
In a group of 10 people, how many ways can we select a committee of 4 people
where 2 must serve as co-chairs?
Solution
Step 1: First, we need to choose 2 people to serve as co-chairs from the 10
people. This can be done in (10
2)ways.
Step 2: Next, we need to choose the remaining 2 people to complete the
committee from the remaining 8 people. This can be done in (8
2)ways.
Step 3: To find the total number of ways to select the committee with 2
co-chairs, we multiply the results from Step 1 and Step 2.
Therefore, the total number of ways to select a committee of 4 people where
2 must serve as co-chairs from a group of 10 people is (10
2)·(8
2)= 45 ·28 = 1260
ways.
8
Question 13
Question
A committee of 5 people is to be formed from a group of 10 women and 8 men.
If the committee must contain at least 2 women, how many different committees
can be formed?
Solution
Step 1: Calculate the total number of ways to form a committee of 5 people
without any restrictions. This is the combination of 18 people taken 5 at a time.
Total ways to form a committee =(18
5)
Step 2: Calculate the number of ways to form a committee with no women.
This is the combination of 8 men taken 5 at a time.
Ways to form a committee with no women =(8
5)
Step 3: Calculate the number of ways to form a committee with exactly 1
woman. This involves choosing 1 woman from 10 women and 4 men from 8
men.
Ways to form a committee with 1 woman =(10
1)×(8
4)
Step 4: Calculate the total number of ways to form a committee with at
least 2 women. This is the total number of ways minus the ways with no women
and with 1 woman.
Ways to form a committee with at least 2 women =(18
5)−(8
5)−((10
1)×(8
4))
Step 5: Substitute the values and calculate the total number of ways.
Ways to form a committee with at least 2 women =(18
5)−(8
5)−((10
1)×(8
4))
Ways to form a committee with at least 2 women =18!
5!(18 −5)!−8!
5!(8 −5)!−(10!
1!(10 −1)! ×8!
4!(8 −4)!)
Ways to form a committee with at least 2 women = 856 −56 −(10 ×70)
Ways to form a committee with at least 2 women = 856 −56 −700
Ways to form a committee with at least 2 women = 100
Therefore, there are 100 different committees that can be formed with at
least 2 women.
9
Question 14
Question
In a group of 10 students, how many ways are there to select a committee of 4
students?
Solution
Step 1: To find the number of ways to select a committee of 4 students from a
group of 10, we will use the combination formula:
C(n, k) = n!
k!(n−k)!
where nis the total number of students and kis the number of students to be
selected for the committee.
Step 2: Substituting n= 10 and k= 4 into the combination formula, we
have:
C(10,4) = 10!
4!(10 −4)!
Step 3: Calculating the factorials:
C(10,4) = 10!
4!6!
C(10,4) = 10 ×9×8×7×6!
4×3×2×1×6!
Step 4: Simplifying the expression:
C(10,4) = 10 ×9×8×7
4×3×2×1
C(10,4) = 210
Step 5: Therefore, there are 210 ways to select a committee of 4 students
from a group of 10 students.
Question 15
Question
A committee of 5 people is to be formed from a group of 10 women and 8 men. If
the committee must consist of at least 3 women, how many different committees
can be formed?
10
Solution
To form a committee of 5 people with at least 3 women, we consider two cases:
1. Exactly 3 women are chosen. 2. All 5 members are women.
Case 1: Exactly 3 women are chosen.
• Selecting 3 women out of 10 can be done in (10
3)ways.
• Selecting 2 men out of 8 can be done in (8
2)ways.
• Using the Multiplication Principle, the total number of committees with
exactly 3 women is (10
3)×(8
2).
Case 2: All 5 members are women.
• Selecting 5 women out of 10 can be done in (10
5)ways.
Total number of committees with at least 3 women:
Total =(10
3)×(8
2)+(10
5)
=10!
3!7! ×8!
2!6! +10!
5!5!
= 120 ×28 + 252
= 3360 + 252
= 3612
Therefore, there are 3612 different committees that can be formed with at
least 3 women.
Question 16
Question
In a group of 10 people, how many ways can 3 people be selected to form a
committee if 2 of the people refuse to be on the committee together?
Solution
Step 1: First, we find the total number of ways to select a committee of 3 people
from a group of 10. Since order doesn’t matter, we will use the combination
formula: (n
r)=n!
r!(n−r)!
where nis the total number of people in the group and ris the number of people
chosen for the committee.
11
Plugging in n= 10 and r= 3:
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120
So, there are 120 ways to select a committee of 3 people from the group.
Step 2: Next, we find the number of ways to select a committee if 2 specific
people refuse to be on the committee together. We will subtract the number of
ways they can be together from the total.
Let’s consider the 2 people who refuse to be on the committee as one unit.
Now we have 9 individuals (8 + the ”unit”) to choose from. To select 3 people
from these 9 individuals, without choosing the ”unit”, we will use the combina-
tion formula again:
(9
3)=9!
3!(9 −3)! =9×8×7
3×2×1= 84
Step 3: Finally, we subtract the number of ways the 2 people can be together
from the total to find the number of ways to form a committee where they are
not together:
120 −84 = 36
So, there are 36 ways to select a committee of 3 people from the group of 10
where the 2 specific people refuse to be on the committee together.
Question 17
Question
In a class of 30 students, 15 are freshmen, 10 are sophomores, and 5 are juniors.
If 8 students are chosen randomly to form a group for a project, what is the
probability that the group will consist of 3 freshmen, 3 sophomores, and 2
juniors?
Solution
Step 1: Find the total number of ways to choose 8 students from a class of 30.
(30
8)=30!
8!(30 −8)! =30!
8!22! = 30,045
Step 2: Find the number of ways to choose 3 freshmen, 3 sophomores, and
2 juniors for the group.
(15
3)×(10
3)×(5
2)=15!
3!(15 −3)!×10!
3!(10 −3)!×5!
2!(5 −2)! = 455×120×10 = 546,000
12
Step 3: Find the probability that the group consists of 3 freshmen, 3 sopho-
mores, and 2 juniors.
P(3 freshmen, 3 sophomores, 2 juniors) = Number of favorable outcomes
Total number of outcomes =546,000
30,045 ≈0.1818
Therefore, the probability that the group will consist of 3 freshmen, 3 sopho-
mores, and 2 juniors is approximately 0.1818.
Question 18
Question
In a mathematics competition, a contestant has to answer 6 out of 10 multiple
choice questions. Each question has 4 possible choices. How many different
ways can the contestant answer the 6 questions if he wants to choose at least
one option from each question?
Solution
Step 1: Determine the number of ways to choose at least one option from each
of the 10 questions.
To ensure that the contestant chooses at least one option from each question,
we can break this down into cases: - Case 1: Exactly one option chosen from
each question. - Case 2: Exactly two options chosen from one question, and one
option chosen from each of the remaining 9 questions. - Case 3: Exactly two
options chosen from two different questions, and one option chosen from each
of the remaining 8 questions. - Case 4: Exactly two options chosen from three
different questions, and one option chosen from each of the remaining 7 ques-
tions. - Case 5: Exactly two options chosen from four different questions, and
one option chosen from each of the remaining 6 questions. - Case 6: One option
chosen from one question, and two options chosen from each of the remaining 5
questions.
We will calculate the number of ways for each case separately.
Step 2: Calculate the number of ways for each case.
- Case 1: 10 ×46ways - Case 2: (10
1)×(9
1)×46ways - Case 3: (10
2)×(8
1)×46
ways - Case 4: (10
3)×(7
1)×46ways - Case 5: (10
4)×(6
1)×46ways - Case 6:
(10
1)×(5
2)×46ways
Step 3: Calculate the total number of ways.
The total number of ways to answer the 6 questions with at least one option
from each question is the sum of the ways from all cases.
Total = Case 1 + Case 2 + Case 3 + Case 4 + Case 5 + Case 6.
13
Question 19
Question
In a group of 10 people, how many different ways can a committee of 3 people
be formed if one person, Alice, must be included?
Solution
Step 1: Since Alice must be included in the committee, we need to choose 2
more people from the remaining 9 people.
Step 2: The number of ways to choose 2 people from a group of 9 is given by
the combination formula (n
r)=n!
r!(n−r)! where nis the total number of people
in the group and ris the number of people to be chosen for the committee.
Step 3: Substituting n= 9 and r= 2 into the formula, we get:
(9
2)=9!
2!(9 −2)! =9!
2!7! =9×8
2×1= 36
Step 4: Therefore, there are 36 different ways to form a committee of 3
people with Alice included from a group of 10 people.
Question 20
Question
In a group of 8 people, how many ways are there to choose a team of 4 people
to represent the group in a competition?
Solution
Step 1: To find the number of ways to choose a team of 4 from a group of 8, we
will use the combination formula. The formula for combination is (n
r)=n!
r!(n−r)! ,
where nis the total number of items and ris the number of items to choose.
Step 2: Substitute n= 8 and r= 4 into the combination formula.
(8
4)=8!
4!(8 −4)!
Step 3: Calculate the factorials in the formula.
8! = 8 ×7×6×5×4×3×2×1 = 40,320
4! = 4 ×3×2×1 = 24
8−4 = 4
4! = 4 ×3×2×1 = 24
14
Step 4: Substitute the factorials back into the formula and simplify.
(8
4)=40,320
24 ×24
(8
4)=40,320
576
(8
4)= 70
Therefore, there are 70 ways to choose a team of 4 people from a group of 8
to represent the group in a competition.
Question 21
Question
In how many ways can a committee of 5 people be formed from a group of 10
men and 8 women if the committee must consist of at least 3 men?
Solution
Step 1: Calculate the number of ways to choose exactly 3, 4, or 5 men.
• Choose 3 men from 10: (10
3)=10!
3!(10−3)! = 120 ways.
• Choose 2 women from 8: (8
2)=8!
2!(8−2)! = 28 ways.
• Total ways to choose 3 men and 2 women: 120 ×28 = 3360 ways.
• Choose 4 men from 10: (10
4)=10!
4!(10−4)! = 210 ways.
• Choose 1 woman from 8: (8
1)=8!
1!(8−1)! = 8 ways.
• Total ways to choose 4 men and 1 woman: 210 ×8 = 1680 ways.
• Choose 5 men from 10: (10
5)=10!
5!(10−5)! = 252 ways.
• Choose 0 women from 8: (8
0)= 1 way.
• Total ways to choose 5 men and 0 women: 252 ×1 = 252 ways.
Step 2: Add up the total number of ways to form the committee with at
least 3 men.
Total ways with at least 3 men = 3360 + 1680 + 252
= 5292 ways
Therefore, there are 5292 ways to form a committee of 5 people from a group
of 10 men and 8 women if the committee must consist of at least 3 men.
15
Question 22
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 women, how many different committees
can be formed?
Solution
Step 1: Find the number of ways to form a committee with exactly 2 women
and 3 men. There are (8
2)ways to choose 2 women from the 8 available, and
(10
3)ways to choose 3 men from the 10 available. Therefore, the number of
committees with exactly 2 women and 3 men is (8
2)×(10
3).
Step 2: Find the number of ways to form a committee with exactly 3 women
and 2 men. Similarly, there are (8
3)ways to choose 3 women and (10
2)ways to
choose 2 men. Therefore, the number of committees with exactly 3 women and
2 men is (8
3)×(10
2).
Step 3: Find the number of ways to form a committee with all 4 or 5 women.
There are (8
4)ways to choose 4 women and (10
1)ways to choose 1 man, and
(8
5)×(10
0)ways to choose all 5 women. Therefore, the number of committees
with all 4 or 5 women is (8
4)×(10
1)+(8
5)×(10
0).
Step 4: Add the results from Step 1, Step 2, and Step 3 to find the total
number of committees. The total number of committees is the sum of commit-
tees with exactly 2 women and 3 men, exactly 3 women and 2 men, and all 4 or
5 women. Therefore, the total number of committees is (8
2)×(10
3)+(8
3)×(10
2)+
((8
4)×(10
1)+(8
5)×(10
0)). Calculate this expression to find the final answer.
Question 23
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 3 men, what is the total number of
different committees that can be formed?
Solution
Step 1: Calculate the total number of ways to form a committee with at least
3 men. Step 2: Calculate the total number of ways to form a committee with
exactly 3 men. Step 3: Calculate the total number of ways to form a committee
with exactly 4 men. Step 4: Calculate the total number of ways to form a
committee with exactly 5 men. Step 5: Add the results from Step 2, Step 3,
and Step 4 to find the total number of different committees that can be formed.
16
Step 1: To calculate the total number of ways to form a committee with at
least 3 men, we can choose 3, 4, or 5 men from the group of 10 men, and then
choose the remaining members of the committee from the group of 8 women.
Step 2: To calculate the total number of ways to form a committee with
exactly 3 men, we choose 3 men from the group of 10 men and choose the
remaining 2 members from the group of 8 women. This can be calculated using
the formula for combinations: (10
3)×(8
2)= 120 ×28 = 3360.
Step 3: To calculate the total number of ways to form a committee with
exactly 4 men, we choose 4 men from the group of 10 men and choose the
remaining member from the group of 8 women. This can be calculated using
the formula for combinations: (10
4)×(8
1)= 210 ×8 = 1680.
Step 4: To calculate the total number of ways to form a committee with
exactly 5 men, we choose all 5 men from the group of 10 men. Since there are
no women on this committee, there is only 1 way to form this committee.
Step 5: Adding the results from Step 2, Step 3, and Step 4: Total number
of different committees that can be formed = 3360 + 1680 + 1 = 5041.
Therefore, the total number of different committees that can be formed from
the group of 10 men and 8 women, with the committee consisting of at least 3
men, is 5041.
Question 24
Question
How many different ways can you arrange the letters in the word ”STATISTICS”
such that no two ’T’s are adjacent?
Solution
To solve this problem, we first find the total number of ways to arrange the
letters in ”STATISTICS”, and then subtract the number of ways where the ’T’s
are adjacent.
Step 1: Calculate the total number of ways to arrange the letters
in ”STATISTICS” The word ”STATISTICS” has 10 letters, but there are
repeating letters: 3 ’S’s, 3 ’T’s, and 2 ’I’s. Thus, the total number of ways to
arrange the letters is given by:
10!
3! ×3! ×2! = 50400
Step 2: Calculate the number of ways the ’T’s are adjacent Consider
the two ’T’s as a single entity (TT). Now we have 9 entities to arrange: TT, S,
S, S, I, I, T, I, C, S. This can be done in:
9!
2! ×3! = 30240
ways.
17
Step 3: Subtract the number of arrangements with ’T’s adjacent
from the total The number of ways to arrange the letters in ”STATISTICS”
such that no two ’T’s are adjacent is:
50400 −30240 = 20160
Therefore, there are 20,160 different ways to arrange the letters in ”STATIS-
TICS” such that no two ’T’s are adjacent.
Question 25
Question
In a group of 10 students, how many ways can we choose 3 students to represent
the group in a competition?
Solution
Step 1: We can solve this problem using the combination formula. The number
of ways to choose kout of nobjects, denoted as C(n, k)or (n
k), is given by:
C(n, k) = n!
k!(n−k)!
Step 2: In this case, we have 10 students and we want to choose 3 to represent
the group. Using the combination formula, we have:
(10
3)=10!
3!(10 −3)!
Step 3: Calculating the factorials, we get:
(10
3)=10 ×9×8
3×2×1
Step 4: Simplifying, we find:
(10
3)= 120
Therefore, there are 120 ways to choose 3 students out of 10 to represent the
group in a competition.
18
Step 4: Calculate the total number of ways to form a committee with 3 men
and 2 women. The total number of ways is given by the product of the number
of ways to choose 3 men and 2 women:
1×21 = 21
Step 5: Determine the probability of selecting a committee with 3 men and
2 women. The probability is the ratio of the number of favorable outcomes to
the total number of outcomes:
P(3 men, 2 women) = 21
252 =1
12
Therefore, the probability that the committee consists of exactly 3 men and
2 women is 1
12 .
Question 2
Question
A committee of 5 people is to be formed from a group of 8 men and 5 women.
If at least 2 men must be on the committee, how many different ways can the
committee be formed?
Solution
Step 1: Find the total number of ways to form a committee with 5 people.
Given that there are 8 men and 5 women, the total number of ways to form a
committee of 5 people from this group is given by the combination formula:
(8+5
5)=(13
5)=13!
5!(13 −5)! = 1287
Step 2: Find the number of ways to form a committee with no men. Since
at least 2 men must be on the committee, find the number of ways to form a
committee with no men first. This will be a committee of 5 women, which is
given by the combination formula:
(5
5)= 1
Step 3: Find the number of ways to form a committee with 1 man. Similarly,
find the number of ways to form a committee with 1 man and 4 women, using
the combination formula:
(8
1)×(5
4)= 8 ×5 = 40
Step 4: Calculate the total number of ways to form a committee with at
least 2 men. To find the number of ways to form a committee with at least 2
2
men, subtract the number of ways with no men and the number of ways with 1
man from the total number of ways:
1287 −1−40 = 1246
Therefore, there are 1246 different ways to form the committee with at least
2 men.
Question 3
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
How many different committees can be formed if the committee must consist of
at least 3 women?
Solution
Step 1: First, we’ll consider the number of ways to choose a committee with
exactly 3 women. Step 2: Select 3 women from the group of 8 women: (8
3)= 56
ways. Step 3: Select the remaining 2 people from the group of 10 men: (10
2)= 45
ways. Step 4: The total number of ways to form a committee with exactly 3
women is 56 ×45 = 2520 ways.
Step 5: Next, we’ll consider the number of ways to choose a committee with
exactly 4 women. Step 6: Select 4 women from the group of 8 women: (8
4)= 70
ways. Step 7: Select the remaining 1 person from the group of 10 men: (10
1)= 10
ways. Step 8: The total number of ways to form a committee with exactly 4
women is 70 ×10 = 700 ways.
Step 9: Finally, we’ll consider the number of ways to choose a committee
with all 5 women. Step 10: Select all 5 women from the group of 8 women:
(8
5)= 56 ways. Step 11: Since there are no men in this case, the remaining 0
people are automatically assigned. Step 12: The total number of ways to form
a committee with all 5 women is 56 ×1 = 56 ways.
Step 13: Therefore, the total number of different committees that can be
formed if the committee must consist of at least 3 women is 2520 + 700 + 56 =
3276 ways.
Question 4
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
How many ways can the committee be formed if it must include at least 3
women?
3
Solution
Step 1: Calculate the number of ways to choose 3 women from the 8 available.
There are (8
3)=8!
3!(8−3)! = 56 ways to choose 3 women from the group.
Step 2: Calculate the number of ways to choose the remaining 2 people from
the remaining pool of 10 men and 5 women (after selecting 3 women). There
are (10+5
2)=(15
2)=15!
2!(15−2)! = 105 ways to choose 2 people from this group.
Step 3: Calculate the total number of ways to form the committee with at
least 3 women. Multiply the results from Step 1 and Step 2 to get the total
number of ways to form the committee: 56 ×105 = 5880 ways.
Therefore, there are 5880 ways to form the committee with at least 3 women
from the group of 10 men and 8 women.
Question 5
Question
A committee of 5 people is to be formed from a group of 8 men and 6 women. If
the committee must consist of at least 2 men and 2 women, how many different
committees can be formed?
Solution
Step 1: Calculate the number of ways to choose 2 men out of 8 men. There are
(8
2)ways to choose 2 men from 8 men. This can be calculated as:
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28
Step 2: Calculate the number of ways to choose 2 women out of 6 women.
There are (6
2)ways to choose 2 women from 6 women. This can be calculated
as: (6
2)=6!
2!(6 −2)! =6×5
2×1= 15
Step 3: Calculate the number of ways to choose the remaining person (either
man or woman). There are 4 remaining people to choose from (6 men and 4
women). Therefore, there are 4 ways to choose the remaining person.
Step 4: Calculate the total number of committees that can be formed. The
total number of committees that can be formed is the product of the number
of ways to choose 2 men, 2 women, and the remaining person. Hence, the total
number of committees is:
28 ×15 ×4 = 1680
Therefore, there are 1680 different committees that can be formed with at
least 2 men and 2 women from the given group of people.
4
Question 6
Question
In a group of 10 people, how many ways can we choose a committee of 4 people
if 2 specific people must be included in the committee?
Solution
Step 1: Choose the 2 specific people who must be included in the committee.
There are (10
2)=10!
2!(10−2)! = 45 ways to choose the 2 specific people.
Step 2: Choose the remaining 2 people to complete the committee. After
selecting the 2 specific people, there are 8 people left to choose from. So, there
are (8
2)=8!
2!(8−2)! = 28 ways to choose the remaining 2 people.
Step 3: Multiply the number of ways from Step 1 and Step 2 to find the
total number of ways to choose the committee. Therefore, the total number
of ways to choose a committee of 4 people with 2 specific people included is
45 ×28 = 1260 ways.
Question 7
Question
A committee of 5 people is to be formed from a group of 12 individuals, including
5 professors and 7 students. How many different committees can be formed if
the committee must contain at least 2 professors?
Solution
Step 1: Find the number of ways to form a committee with exactly 2 professors.
There are 5ways to choose the first professor and 4ways to choose the second
professor. For the remaining 3spots, we can choose from the remaining 12−5 =
7individuals. Therefore, the number of ways to form a committee with exactly
2 professors is:
5×4×(7
3)= 5 ×4×7!
3!4! = 5 ×4×35 = 700
Step 2: Find the number of ways to form a committee with exactly 3 profes-
sors. There are (5
3)=(5
2)= 10 ways to choose 3 professors from the 5 available.
For the remaining 2 spots, we can choose from the 7 students. Therefore, the
number of ways to form a committee with exactly 3 professors is:
10 ×(7
2)= 10 ×7!
2!5! = 10 ×21 = 210
Step 3: Find the number of ways to form a committee with exactly 4 pro-
fessors. There are (5
4)= 5 ways to choose 4 professors from the 5 available. For
5
the remaining spot, we can choose from the 7 students. Therefore, the number
of ways to form a committee with exactly 4 professors is:
5×7 = 35
Step 4: Find the number of ways to form a committee with all 5 professors.
There is only (5
5)= 1 way to choose all 5 professors.
Step 5: Add up the results from Step 1, Step 2, Step 3, and Step 4 to get
the total number of ways to form a committee with at least 2 professors:
700 + 210 + 35 + 1 = 946
Question 8
Question
In a group of 10 students, how many ways can you choose a committee of 4
students if 2 of them are best friends and must be on the committee together?
Solution
Step 1: Choose the two best friends to be on the committee. There is only 1
way to choose the best friends to be on the committee together.
Step 2: Choose the remaining 2 students to fill the committee. Since we
have already chosen the two best friends, we need to choose 2 students from the
remaining 8 students. The number of ways to choose 2 students from 8 is given
by the combination formula:
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28
Step 3: Multiply the results from Step 1 and Step 2 to find the total number
of ways to choose the committee. The total number of ways to choose the
committee with the two best friends is:
1×28 = 28
Therefore, there are 28 ways to choose a committee of 4 students from a
group of 10 students where 2 of them are best friends and must be on the
committee together.
Question 9
Question
A committee of 5 people is to be formed from a group of 10 men and 8 women.
In how many ways can this committee be formed if it must consist of 3 men and
2 women?
6
Solution
Step 1: Calculate the number of ways to choose 3 men out of 10. This can be
done using combinations.
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120
Step 2: Calculate the number of ways to choose 2 women out of 8. This can
also be done using combinations.
(8
2)=8!
2!(8 −2)! =8×7
2×1= 28
Step 3: Multiply the results from Step 1 and Step 2 to find the total number
of ways to form the committee. Total number of ways = 120 ×28 = 3360
Therefore, there are 3360 ways to form a committee of 3 men and 2 women
from a group of 10 men and 8 women.
Question 10
Question
A research team is composing a committee of 5 members from a pool of 12
researchers. If 3 of the researchers are statisticians and 9 are not, how many
different committees can be formed if each committee must include at least 2
statisticians?
Solution
Step 1: Calculate the number of committees with exactly 2 statisticians: Since
there are 3 statisticians, the number of ways to choose 2 of them is (3
2)= 3. The
remaining 3 members must be chosen from the 9 non-statisticians. This can be
done in (9
3)= 84 ways. Therefore, the total number of committees with exactly
2 statisticians is 3×84 = 252.
Step 2: Calculate the number of committees with exactly 3 statisticians:
Similarly, the number of ways to choose 3 statisticians from the pool of 3 is
(3
3)= 1. The remaining 2 members must be chosen from the 9 non-statisticians.
This can be done in (9
2)= 36 ways. Therefore, the total number of committees
with exactly 3 statisticians is 1×36 = 36.
Step 3: Calculate the number of committees with 4 or 5 statisticians: Since
there are not enough statisticians left to form committees with 4 or 5 statisti-
cians, the number of such committees is 0.
Step 4: Calculate the total number of committees with at least 2 statisticians:
The total number of committees with at least 2 statisticians is the sum of the
committees with exactly 2 and exactly 3 statisticians, which is 252 + 36 = 288.
Therefore, there are 288 different committees that can be formed from a pool
of 12 researchers such that each committee must include at least 2 statisticians.
7
Question 11
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 form a committee with exactly 3
women: There are (8
3)ways to select 3 women from 8, and (10
2)ways to select
2 men from 10. Therefore, the number of ways to form the committee with
exactly 3 women is: (8
3)×(10
2)
Step 2: Calculate the number of ways to form a committee with exactly 4
women: There are (8
4)ways to select 4 women from 8, and (10
1)ways to select
1 man from 10. Therefore, the number of ways to form the committee with
exactly 4 women is: (8
4)×(10
1)
Step 3: Calculate the number of ways to form a committee with exactly 5
women: There is only (8
5)way to select 5 women from 8. Therefore, the number
of ways to form the committee with exactly 5 women is: (8
5)
Step 4: Add up the results from Step 1, Step 2, and Step 3 to get the
total number of ways to form a committee with at least 3 women: Total =
(8
3)×(10
2)+(8
4)×(10
1)+(8
5)
Calculating this expression will give us the final answer.
Question 12
Question
In a group of 10 people, how many ways can we select a committee of 4 people
where 2 must serve as co-chairs?
Solution
Step 1: First, we need to choose 2 people to serve as co-chairs from the 10
people. This can be done in (10
2)ways.
Step 2: Next, we need to choose the remaining 2 people to complete the
committee from the remaining 8 people. This can be done in (8
2)ways.
Step 3: To find the total number of ways to select the committee with 2
co-chairs, we multiply the results from Step 1 and Step 2.
Therefore, the total number of ways to select a committee of 4 people where
2 must serve as co-chairs from a group of 10 people is (10
2)·(8
2)= 45 ·28 = 1260
ways.
8
Question 13
Question
A committee of 5 people is to be formed from a group of 10 women and 8 men.
If the committee must contain at least 2 women, how many different committees
can be formed?
Solution
Step 1: Calculate the total number of ways to form a committee of 5 people
without any restrictions. This is the combination of 18 people taken 5 at a time.
Total ways to form a committee =(18
5)
Step 2: Calculate the number of ways to form a committee with no women.
This is the combination of 8 men taken 5 at a time.
Ways to form a committee with no women =(8
5)
Step 3: Calculate the number of ways to form a committee with exactly 1
woman. This involves choosing 1 woman from 10 women and 4 men from 8
men.
Ways to form a committee with 1 woman =(10
1)×(8
4)
Step 4: Calculate the total number of ways to form a committee with at
least 2 women. This is the total number of ways minus the ways with no women
and with 1 woman.
Ways to form a committee with at least 2 women =(18
5)−(8
5)−((10
1)×(8
4))
Step 5: Substitute the values and calculate the total number of ways.
Ways to form a committee with at least 2 women =(18
5)−(8
5)−((10
1)×(8
4))
Ways to form a committee with at least 2 women =18!
5!(18 −5)!−8!
5!(8 −5)!−(10!
1!(10 −1)! ×8!
4!(8 −4)!)
Ways to form a committee with at least 2 women = 856 −56 −(10 ×70)
Ways to form a committee with at least 2 women = 856 −56 −700
Ways to form a committee with at least 2 women = 100
Therefore, there are 100 different committees that can be formed with at
least 2 women.
9
Question 14
Question
In a group of 10 students, how many ways are there to select a committee of 4
students?
Solution
Step 1: To find the number of ways to select a committee of 4 students from a
group of 10, we will use the combination formula:
C(n, k) = n!
k!(n−k)!
where nis the total number of students and kis the number of students to be
selected for the committee.
Step 2: Substituting n= 10 and k= 4 into the combination formula, we
have:
C(10,4) = 10!
4!(10 −4)!
Step 3: Calculating the factorials:
C(10,4) = 10!
4!6!
C(10,4) = 10 ×9×8×7×6!
4×3×2×1×6!
Step 4: Simplifying the expression:
C(10,4) = 10 ×9×8×7
4×3×2×1
C(10,4) = 210
Step 5: Therefore, there are 210 ways to select a committee of 4 students
from a group of 10 students.
Question 15
Question
A committee of 5 people is to be formed from a group of 10 women and 8 men. If
the committee must consist of at least 3 women, how many different committees
can be formed?
10
Solution
To form a committee of 5 people with at least 3 women, we consider two cases:
1. Exactly 3 women are chosen. 2. All 5 members are women.
Case 1: Exactly 3 women are chosen.
• Selecting 3 women out of 10 can be done in (10
3)ways.
• Selecting 2 men out of 8 can be done in (8
2)ways.
• Using the Multiplication Principle, the total number of committees with
exactly 3 women is (10
3)×(8
2).
Case 2: All 5 members are women.
• Selecting 5 women out of 10 can be done in (10
5)ways.
Total number of committees with at least 3 women:
Total =(10
3)×(8
2)+(10
5)
=10!
3!7! ×8!
2!6! +10!
5!5!
= 120 ×28 + 252
= 3360 + 252
= 3612
Therefore, there are 3612 different committees that can be formed with at
least 3 women.
Question 16
Question
In a group of 10 people, how many ways can 3 people be selected to form a
committee if 2 of the people refuse to be on the committee together?
Solution
Step 1: First, we find the total number of ways to select a committee of 3 people
from a group of 10. Since order doesn’t matter, we will use the combination
formula: (n
r)=n!
r!(n−r)!
where nis the total number of people in the group and ris the number of people
chosen for the committee.
11
Plugging in n= 10 and r= 3:
(10
3)=10!
3!(10 −3)! =10 ×9×8
3×2×1= 120
So, there are 120 ways to select a committee of 3 people from the group.
Step 2: Next, we find the number of ways to select a committee if 2 specific
people refuse to be on the committee together. We will subtract the number of
ways they can be together from the total.
Let’s consider the 2 people who refuse to be on the committee as one unit.
Now we have 9 individuals (8 + the ”unit”) to choose from. To select 3 people
from these 9 individuals, without choosing the ”unit”, we will use the combina-
tion formula again:
(9
3)=9!
3!(9 −3)! =9×8×7
3×2×1= 84
Step 3: Finally, we subtract the number of ways the 2 people can be together
from the total to find the number of ways to form a committee where they are
not together:
120 −84 = 36
So, there are 36 ways to select a committee of 3 people from the group of 10
where the 2 specific people refuse to be on the committee together.
Question 17
Question
In a class of 30 students, 15 are freshmen, 10 are sophomores, and 5 are juniors.
If 8 students are chosen randomly to form a group for a project, what is the
probability that the group will consist of 3 freshmen, 3 sophomores, and 2
juniors?
Solution
Step 1: Find the total number of ways to choose 8 students from a class of 30.
(30
8)=30!
8!(30 −8)! =30!
8!22! = 30,045
Step 2: Find the number of ways to choose 3 freshmen, 3 sophomores, and
2 juniors for the group.
(15
3)×(10
3)×(5
2)=15!
3!(15 −3)!×10!
3!(10 −3)!×5!
2!(5 −2)! = 455×120×10 = 546,000
12
Step 3: Find the probability that the group consists of 3 freshmen, 3 sopho-
mores, and 2 juniors.
P(3 freshmen, 3 sophomores, 2 juniors) = Number of favorable outcomes
Total number of outcomes =546,000
30,045 ≈0.1818
Therefore, the probability that the group will consist of 3 freshmen, 3 sopho-
mores, and 2 juniors is approximately 0.1818.
Question 18
Question
In a mathematics competition, a contestant has to answer 6 out of 10 multiple
choice questions. Each question has 4 possible choices. How many different
ways can the contestant answer the 6 questions if he wants to choose at least
one option from each question?
Solution
Step 1: Determine the number of ways to choose at least one option from each
of the 10 questions.
To ensure that the contestant chooses at least one option from each question,
we can break this down into cases: - Case 1: Exactly one option chosen from
each question. - Case 2: Exactly two options chosen from one question, and one
option chosen from each of the remaining 9 questions. - Case 3: Exactly two
options chosen from two different questions, and one option chosen from each
of the remaining 8 questions. - Case 4: Exactly two options chosen from three
different questions, and one option chosen from each of the remaining 7 ques-
tions. - Case 5: Exactly two options chosen from four different questions, and
one option chosen from each of the remaining 6 questions. - Case 6: One option
chosen from one question, and two options chosen from each of the remaining 5
questions.
We will calculate the number of ways for each case separately.
Step 2: Calculate the number of ways for each case.
- Case 1: 10 ×46ways - Case 2: (10
1)×(9
1)×46ways - Case 3: (10
2)×(8
1)×46
ways - Case 4: (10
3)×(7
1)×46ways - Case 5: (10
4)×(6
1)×46ways - Case 6:
(10
1)×(5
2)×46ways
Step 3: Calculate the total number of ways.
The total number of ways to answer the 6 questions with at least one option
from each question is the sum of the ways from all cases.
Total = Case 1 + Case 2 + Case 3 + Case 4 + Case 5 + Case 6.
13
Question 19
Question
In a group of 10 people, how many different ways can a committee of 3 people
be formed if one person, Alice, must be included?
Solution
Step 1: Since Alice must be included in the committee, we need to choose 2
more people from the remaining 9 people.
Step 2: The number of ways to choose 2 people from a group of 9 is given by
the combination formula (n
r)=n!
r!(n−r)! where nis the total number of people
in the group and ris the number of people to be chosen for the committee.
Step 3: Substituting n= 9 and r= 2 into the formula, we get:
(9
2)=9!
2!(9 −2)! =9!
2!7! =9×8
2×1= 36
Step 4: Therefore, there are 36 different ways to form a committee of 3
people with Alice included from a group of 10 people.
Question 20
Question
In a group of 8 people, how many ways are there to choose a team of 4 people
to represent the group in a competition?
Solution
Step 1: To find the number of ways to choose a team of 4 from a group of 8, we
will use the combination formula. The formula for combination is (n
r)=n!
r!(n−r)! ,
where nis the total number of items and ris the number of items to choose.
Step 2: Substitute n= 8 and r= 4 into the combination formula.
(8
4)=8!
4!(8 −4)!
Step 3: Calculate the factorials in the formula.
8! = 8 ×7×6×5×4×3×2×1 = 40,320
4! = 4 ×3×2×1 = 24
8−4 = 4
4! = 4 ×3×2×1 = 24
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Step 4: Substitute the factorials back into the formula and simplify.
(8
4)=40,320
24 ×24
(8
4)=40,320
576
(8
4)= 70
Therefore, there are 70 ways to choose a team of 4 people from a group of 8
to represent the group in a competition.
Question 21
Question
In how many ways can a committee of 5 people be formed from a group of 10
men and 8 women if the committee must consist of at least 3 men?
Solution
Step 1: Calculate the number of ways to choose exactly 3, 4, or 5 men.
• Choose 3 men from 10: (10
3)=10!
3!(10−3)! = 120 ways.
• Choose 2 women from 8: (8
2)=8!
2!(8−2)! = 28 ways.
• Total ways to choose 3 men and 2 women: 120 ×28 = 3360 ways.
• Choose 4 men from 10: (10
4)=10!
4!(10−4)! = 210 ways.
• Choose 1 woman from 8: (8
1)=8!
1!(8−1)! = 8 ways.
• Total ways to choose 4 men and 1 woman: 210 ×8 = 1680 ways.
• Choose 5 men from 10: (10
5)=10!
5!(10−5)! = 252 ways.
• Choose 0 women from 8: (8
0)= 1 way.
• Total ways to choose 5 men and 0 women: 252 ×1 = 252 ways.
Step 2: Add up the total number of ways to form the committee with at
least 3 men.
Total ways with at least 3 men = 3360 + 1680 + 252
= 5292 ways
Therefore, there are 5292 ways to form a committee of 5 people from a group
of 10 men and 8 women if the committee must consist of at least 3 men.
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Question 22
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 women, how many different committees
can be formed?
Solution
Step 1: Find the number of ways to form a committee with exactly 2 women
and 3 men. There are (8
2)ways to choose 2 women from the 8 available, and
(10
3)ways to choose 3 men from the 10 available. Therefore, the number of
committees with exactly 2 women and 3 men is (8
2)×(10
3).
Step 2: Find the number of ways to form a committee with exactly 3 women
and 2 men. Similarly, there are (8
3)ways to choose 3 women and (10
2)ways to
choose 2 men. Therefore, the number of committees with exactly 3 women and
2 men is (8
3)×(10
2).
Step 3: Find the number of ways to form a committee with all 4 or 5 women.
There are (8
4)ways to choose 4 women and (10
1)ways to choose 1 man, and
(8
5)×(10
0)ways to choose all 5 women. Therefore, the number of committees
with all 4 or 5 women is (8
4)×(10
1)+(8
5)×(10
0).
Step 4: Add the results from Step 1, Step 2, and Step 3 to find the total
number of committees. The total number of committees is the sum of commit-
tees with exactly 2 women and 3 men, exactly 3 women and 2 men, and all 4 or
5 women. Therefore, the total number of committees is (8
2)×(10
3)+(8
3)×(10
2)+
((8
4)×(10
1)+(8
5)×(10
0)). Calculate this expression to find the final answer.
Question 23
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 3 men, what is the total number of
different committees that can be formed?
Solution
Step 1: Calculate the total number of ways to form a committee with at least
3 men. Step 2: Calculate the total number of ways to form a committee with
exactly 3 men. Step 3: Calculate the total number of ways to form a committee
with exactly 4 men. Step 4: Calculate the total number of ways to form a
committee with exactly 5 men. Step 5: Add the results from Step 2, Step 3,
and Step 4 to find the total number of different committees that can be formed.
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Step 1: To calculate the total number of ways to form a committee with at
least 3 men, we can choose 3, 4, or 5 men from the group of 10 men, and then
choose the remaining members of the committee from the group of 8 women.
Step 2: To calculate the total number of ways to form a committee with
exactly 3 men, we choose 3 men from the group of 10 men and choose the
remaining 2 members from the group of 8 women. This can be calculated using
the formula for combinations: (10
3)×(8
2)= 120 ×28 = 3360.
Step 3: To calculate the total number of ways to form a committee with
exactly 4 men, we choose 4 men from the group of 10 men and choose the
remaining member from the group of 8 women. This can be calculated using
the formula for combinations: (10
4)×(8
1)= 210 ×8 = 1680.
Step 4: To calculate the total number of ways to form a committee with
exactly 5 men, we choose all 5 men from the group of 10 men. Since there are
no women on this committee, there is only 1 way to form this committee.
Step 5: Adding the results from Step 2, Step 3, and Step 4: Total number
of different committees that can be formed = 3360 + 1680 + 1 = 5041.
Therefore, the total number of different committees that can be formed from
the group of 10 men and 8 women, with the committee consisting of at least 3
men, is 5041.
Question 24
Question
How many different ways can you arrange the letters in the word ”STATISTICS”
such that no two ’T’s are adjacent?
Solution
To solve this problem, we first find the total number of ways to arrange the
letters in ”STATISTICS”, and then subtract the number of ways where the ’T’s
are adjacent.
Step 1: Calculate the total number of ways to arrange the letters
in ”STATISTICS” The word ”STATISTICS” has 10 letters, but there are
repeating letters: 3 ’S’s, 3 ’T’s, and 2 ’I’s. Thus, the total number of ways to
arrange the letters is given by:
10!
3! ×3! ×2! = 50400
Step 2: Calculate the number of ways the ’T’s are adjacent Consider
the two ’T’s as a single entity (TT). Now we have 9 entities to arrange: TT, S,
S, S, I, I, T, I, C, S. This can be done in:
9!
2! ×3! = 30240
ways.
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Step 3: Subtract the number of arrangements with ’T’s adjacent
from the total The number of ways to arrange the letters in ”STATISTICS”
such that no two ’T’s are adjacent is:
50400 −30240 = 20160
Therefore, there are 20,160 different ways to arrange the letters in ”STATIS-
TICS” such that no two ’T’s are adjacent.
Question 25
Question
In a group of 10 students, how many ways can we choose 3 students to represent
the group in a competition?
Solution
Step 1: We can solve this problem using the combination formula. The number
of ways to choose kout of nobjects, denoted as C(n, k)or (n
k), is given by:
C(n, k) = n!
k!(n−k)!
Step 2: In this case, we have 10 students and we want to choose 3 to represent
the group. Using the combination formula, we have:
(10
3)=10!
3!(10 −3)!
Step 3: Calculating the factorials, we get:
(10
3)=10 ×9×8
3×2×1
Step 4: Simplifying, we find:
(10
3)= 120
Therefore, there are 120 ways to choose 3 students out of 10 to represent the
group in a competition.
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