Combinations in Selecting a Group of Students: A Comprehensive Guide

When dealing with problems involving combinations, it's essential to understand the underlying principles and how to apply them effectively. This article explores a specific problem where a batch of 5 boys and 6 girls needs to be combined into groups of 5 students in different formations. We will delve into the mathematical calculations and explore the logic behind solving such problems.

Problem Description

Given a pool of students consisting of 5 boys and 6 girls, we need to determine the number of ways to form groups of 5 students in two different ways:

3 boys and 2 girls 2 boys and 3 girls

Step-by-Step Calculation

Selection of 3 Boys and 2 Girls

First, let's calculate the number of ways to select 3 boys out of 5:

{{5 choose 3}} frac{5!}{3!(5-3)!} frac{5 times 4 times 3!}{3! times 2!} 10

Next, let's calculate the number of ways to select 2 girls out of 6:

{{6 choose 2}} frac{6!}{2!(6-2)!} frac{6 times 5 times 4!}{2! times 4!} 15

Combining these, the number of ways to form a group of 3 boys and 2 girls is:

{{5 choose 3}} times {{6 choose 2}} 10 times 15 150

Therefore, there are 150 combinations to form a group of 3 boys and 2 girls.

Selection of 2 Boys and 3 Girls

Now, let's calculate the number of ways to select 2 boys out of 5:

{{5 choose 2}} frac{5!}{2!(5-2)!} frac{5 times 4 times 3!}{2! times 3!} 10

Next, let's calculate the number of ways to select 3 girls out of 6:

{{6 choose 3}} frac{6!}{3!(6-3)!} frac{6 times 5 times 4 times 3!}{3! times 3! times 2 times 1} 20

Combining these, the number of ways to form a group of 2 boys and 3 girls is:

{{5 choose 2}} times {{6 choose 3}} 10 times 20 200

Therefore, there are 200 combinations to form a group of 2 boys and 3 girls.

Total Combinations

To find the total number of combinations for both group formations, we simply add the results from the previous calculations:

Total combinations 150 (3 boys and 2 girls) 200 (2 boys and 3 girls) 350

Hence, there are a total of 350 combinations to form groups of 5 students with the specified configurations.

Conclusion

Solving combination problems involves understanding the principles of binomial coefficients and how to apply them effectively. By breaking down the problem into smaller parts, we can easily calculate the number of ways to form different groups of students.

The key takeaways from this problem are:

Understanding and applying binomial coefficients Breaking down the problem into simpler parts Adding the individual results to get the total combinations

These skills are not only useful in this specific problem but also in a broader range of mathematical and real-world applications.