Forming a Committee: A Detailed Analysis of Combinatorial Choices

When organizing a committee, the number of ways to form the committee from a pool of individuals involves combinatorial mathematics. Specifically, this article will explore the problem of forming a committee of 5 members from a group of 5 men and 8 women, totaling 13 individuals. We will use the combination formula and combinatorial principles to calculate the number of possible ways to form such a committee.

Basic Combination Formula

The combination formula is given by:

(binom{n}{r} frac{n!}{r!(n-r)!})

where: n is the total number of people to choose from. r is the number of people to choose. n! denotes the factorial of n, which is the product of all positive integers up to that number.

Application to the Committee Formation

In this scenario, (n 13), representing 5 men and 8 women, and (r 5) for the committee members. Substituting these values into the combination formula:

(binom{13}{5} frac{13!}{5! cdot 8!})

To calculate this, we break it down step by step:

13! 13 times 12 times 11 times 10 times 9 times 8!

The numerator simplifies to:

13 times 12 156 156 times 11 1716 1716 times 10 17160 17160 times 9 154440

The denominator is:

5! 120

Therefore:

(binom{13}{5} frac{154440}{120} 1287)

This means there are 1287 ways to form the committee.

Subsets of the Committee with Different Gender Combinations

The committee can have 0, 1, 2, 3, or 4 men. For each of the possible choices of k men, there are combinations of women from the remaining pool. The formula can be expressed as:

(sum_{k0}^4 binom{4}{k} binom{6}{5-k} binom{10}{5})

Using this, we can calculate the different combinations:

2W3M: (binom{4}{2} binom{6}{3} 6 times 20 120) 3W2M: (binom{4}{3} binom{6}{2} 4 times 15 60) 4W1M: (binom{4}{4} binom{6}{1} 1 times 6 6) 5W: (binom{4}{5} 0) (impossible) 0M: (binom{4}{0} binom{6}{5} 1 times 6 6)

Adding these together:

120 60 6 0 6 192

Alternative Calculation Method

Another method involves directly calculating the total number of ways to choose 5 people from 13:

(binom{13}{5} frac{13!}{5!8!} 1287)

Subtracting those configurations that have no women (i.e., choosing 5 men from 5) or only one woman (i.e., choosing 4 men from 5 and 1 woman from 8):

(binom{5}{5} 1) (binom{5}{4} binom{8}{1} 5 times 8 40)

This leaves:

1287 - 1 - 40 1246

Conclusion

Thus, we have analyzed the problem from multiple angles, first by calculating the total number of ways to form a committee and then by considering the specific gender compositions. The correct number of ways to form the committee is 1287, with detailed breakdowns of possible combinations.