Combinations and Group Formation: How Many Groups of 2 Women and 2 Men Can Be Formed?

Combinatorics is a fascinating branch of mathematics that deals with the study of finite or countable discrete structures. One of the fundamental concepts in combinatorics is the idea of combinations, which is used to determine the number of ways elements can be selected from a set without regard to the order of selection. In this article, we will explore how to determine the number of groups that can be formed consisting of exactly 2 women and 2 men from a pool of 8 men and 5 women.

The Problem and Solution

The task at hand is to find out how many groups can be formed with exactly 2 women and 2 men from a group of 8 men and 5 women. To solve this, we will use the concept of combinations, which is a specific branch of combinatorics.

Step 1: Choosing 2 Women from 5

First, we need to calculate the number of ways to choose 2 women from a group of 5. This can be represented using the combination formula:

C(n, k) frac{n!}{k!(n-k)!}

Here, n is the total number of elements (5 women), and k is the number of elements to be chosen (2 women).

C(5, 2) frac{5!}{2!(5-2)!} frac{5 times 4}{2 times 1} 10

Step 2: Choosing 2 Men from 8

Next, we calculate the number of ways to choose 2 men from a group of 8. Using the same combination formula:

C(8, 2) frac{8!}{2!(8-2)!} frac{8 times 7}{2 times 1} 28

Step 3: Combining the Results

To find the total number of groups, we multiply the number of ways to choose the women by the number of ways to choose the men:

Total groups C(5, 2) times C(8, 2) 10 times 28 280

Therefore, the total number of groups that can be formed with exactly 2 women and 2 men is 280.

Explanation and Insights

This problem is a classic example of combinations, which is crucial in various fields such as probability, statistics, and computer science. Understanding how to calculate combinations is essential because it helps in solving a wide range of real-world problems, from selecting teams to choosing items in a store.

Conclusion

Through the method of combinations, we have determined that 280 distinct groups can be formed from 8 men and 5 women with exactly 2 women and 2 men in each group. This solution is derived using the fundamental principles of combinatorics and highlights the elegance and power of mathematical problem-solving techniques.