Introduction to Combinatorics and Fundraising Committees

In the context of combinatorics, the problem involves selecting a committee of 5 people from a pool of 7 men and 8 women. This is a fundamental exercise in combinatorial mathematics, where we determine the number of distinct ways a specific group of people can be chosen from a larger set. The goal is to understand and apply the concept of combinations to real-world scenarios such as forming a fundraising committee.

Concept of Combinations

The core of this problem is the use of combinations, which are a way of selecting items from a larger set without regard to the order of selection. The formula for combinations is given by:

Cnk n! / (k! (n-k)!)

where:

n is the total number of items to choose from, k is the number of items to be chosen, n! is the factorial of n.

Application to the Fundraising Committee

In this specific problem, we have:

Total number of men and women (people): 7 men 8 women 15 people, Number of people to be selected for the committee: 5 people.

Plugging these numbers into the combination formula, we get:

C155 15! / (5! (15-5)!) 15! / (5! 10!)

Calculating the factorials:

15! 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

5! 5 x 4 x 3 x 2 x 1

10! 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

Simplifying the terms, we can cancel out the common factors in the numerator and the denominator:

15! / (5! 10!) (15 x 14 x 13 x 12 x 11) / (1 x 1 x 1 x 1 x 1) 3003

Therefore, there are 3003 distinct ways to form a fundraising committee of 5 people from a group of 7 men and 8 women.

Further Explorations of Combinatorial Scenarios

When the structure of the committee is further specified, such as requiring a certain number of men and women, the problem can be broken down into specific combinations. For example:

2 women and 3 men 3 women and 2 men 4 women and 1 man 5 women and no men

The number of ways to form such committees can be calculated by:

Calculating the combinations for the specified gender distribution. Summing up the results for all valid distributions.

For instance, the number of ways to form a 2W3M committee is:

C82 * C73 (8! / (2! 6!)) * (7! / (3! 4!)) 28 * 35 980

Conclusion

Understanding the concept of combinations is crucial in solving real-world problems like forming a fundraising committee. By breaking down the problem into specific scenarios, we can apply combinatorial methods to find the number of distinct ways to form the committee. This approach not only helps in organizing events but also in making informed decisions in various fields that involve group selection.