Calculating the Number of Ways to Select Girls

Oct 13, 2025 by windailys.com 32 views

Introduction

In this article, we will explore the concept of forming a committee from a group of students. We have 6 boys and 5 girls in a class, and we want to form a committee of 5 members such that it includes 2 girls and 3 boys. This is a classic problem of combinatorics, and we will use mathematical techniques to find the number of ways to form such a committee.

Understanding the Problem

To form a committee of 5 members with 2 girls and 3 boys, we need to select 2 girls from a group of 5 girls and 3 boys from a group of 6 boys. This is a combination problem, as the order in which we select the members does not matter.

Calculating the Number of Ways to Select Girls

First, let's calculate the number of ways to select 2 girls from a group of 5 girls. This can be done using the combination formula:

C(n, k) = n! / (k!(n-k)!)

where n is the total number of items, k is the number of items to select, and ! denotes the factorial function.

In this case, n = 5 (the total number of girls) and k = 2 (the number of girls to select). Plugging these values into the formula, we get:

C(5, 2) = 5! / (2!(5-2)!) = 5! / (2!3!) = (5 × 4 × 3 × 2 × 1) / ((2 × 1)(3 × 2 × 1)) = 120 / (2 × 6) = 120 / 12 = 10

So, there are 10 ways to select 2 girls from a group of 5 girls.

Calculating the Number of Ways to Select Boys

Next, let's calculate the number of ways to select 3 boys from a group of 6 boys. Again, we use the combination formula:

C(n, k) = n! / (k!(n-k)!)

In this case, n = 6 (the total number of boys) and k = 3 (the number of boys to select). Plugging these values into the formula, we get:

C(6, 3) = 6! / (3!(6-3)!) = 6! / (3!3!) = (6 × 5 × 4 × 3 × 2 × 1) / ((3 × 2 × 1)(3 × 2 × 1)) = 720 / (6 × 6) = 720 / 36 = 20

So, there are 20 ways to select 3 boys from a group of 6 boys.

Calculating the Total Number of Ways to Form the Committee

Now that we have calculated the number of ways to select 2 girls and 3 boys separately, we can find the total number of ways to form the committee by multiplying these two numbers together:

Total number of ways = Number of ways to select girls × Number of ways to select boys = 10 × 20 = 200

Therefore, there are 200 ways to form a committee of 5 members with 2 girls and 3 boys.

Conclusion

In this article, we used mathematical techniques to find the number of ways to form a committee of 5 members with 2 girls and 3 boys. We calculated the number of ways to select 2 girls from a group of 5 girls and 3 boys from a group of 6 boys separately, and then multiplied these two numbers together to find the total number of ways to form the committee. The result is 200 ways, which is the final answer to the problem.

References

"Combinatorics: Topics, Techniques, Algorithms" by Peter J. Cameron
"Introduction to Combinatorics" by Richard P. Stanley

Q: What is the formula for calculating the number of ways to select a committee?

A: The formula for calculating the number of ways to select a committee is:

C(n, k) = n! / (k!(n-k)!)

where n is the total number of items, k is the number of items to select, and ! denotes the factorial function.

Q: How do I calculate the number of ways to select 2 girls from a group of 5 girls?

A: To calculate the number of ways to select 2 girls from a group of 5 girls, you can use the combination formula:

C(5, 2) = 5! / (2!(5-2)!) = 5! / (2!3!) = (5 × 4 × 3 × 2 × 1) / ((2 × 1)(3 × 2 × 1)) = 120 / (2 × 6) = 120 / 12 = 10

So, there are 10 ways to select 2 girls from a group of 5 girls.

Q: How do I calculate the number of ways to select 3 boys from a group of 6 boys?

A: To calculate the number of ways to select 3 boys from a group of 6 boys, you can use the combination formula:

C(6, 3) = 6! / (3!(6-3)!) = 6! / (3!3!) = (6 × 5 × 4 × 3 × 2 × 1) / ((3 × 2 × 1)(3 × 2 × 1)) = 720 / (6 × 6) = 720 / 36 = 20

So, there are 20 ways to select 3 boys from a group of 6 boys.

Q: How do I find the total number of ways to form the committee?

A: To find the total number of ways to form the committee, you can multiply the number of ways to select 2 girls and 3 boys together:

Total number of ways = Number of ways to select girls × Number of ways to select boys = 10 × 20 = 200

Therefore, there are 200 ways to form a committee of 5 members with 2 girls and 3 boys.

Q: What if I want to select 3 girls and 2 boys instead?

A: If you want to select 3 girls and 2 boys, you can use the combination formula to calculate the number of ways to select 3 girls from a group of 5 girls and 2 boys from a group of 6 boys:

C(5, 3) = 5! / (3!(5-3)!) = 5! / (3!2!) = (5 × 4 × 3 × 2 × 1) / ((3 × 2 × 1)(2 × 1)) = 120 / (6 × 2) = 120 / 12 = 10

C(6, 2) = 6! / (2!(6-2)!) = 6! / (2!4!) = (6 × 5 × 4 × 3 × 2 × 1) / ((2 × 1)(4 × 3 × 2 × 1)) = 720 / (2 × 24) = 720 / 48 = 15

Total number of ways = Number of ways to select girls × Number of ways to select boys = 10 × 15 = 150

Therefore, there are 150 ways to form a committee of 5 members with 3 girls and 2 boys.

Q: What if I want to select 4 girls and 1 boy instead?

A: If you want to select 4 girls and 1 boy, you can use the combination formula to calculate the number of ways to select 4 girls from a group of 5 girls and 1 boy from a group of 6 boys:

C(5, 4) = 5! / (4!(5-4)!) = 5! / (4!1!) = (5 × 4 × 3 × 2 × 1) / ((4 × 3 × 2 × 1)(1 × 1)) = 120 / (24 × 1) = 120 / 24 = 5

C(6, 1) = 6! / (1!(6-1)!) = 6! / (1!5!) = (6 × 5 × 4 × 3 × 2 × 1) / ((1 × 1)(5 × 4 × 3 × 2 × 1)) = 720 / (1 × 120) = 720 / 120 = 6

Total number of ways = Number of ways to select girls × Number of ways to select boys = 5 × 6 = 30

Therefore, there are 30 ways to form a committee of 5 members with 4 girls and 1 boy.

Conclusion

In this article, we have answered some frequently asked questions about forming a committee. We have used the combination formula to calculate the number of ways to select 2 girls and 3 boys, 3 girls and 2 boys, 4 girls and 1 boy, and other combinations. We have also provided examples of how to use the combination formula to solve these problems.