Combinations and Committees: Mathematical and Practical Approaches

Understanding combinations and how to form committees under specific conditions is a fundamental concept in combinatorics. In this article, we will delve into several scenarios involving the creation of committees and how to calculate the number of ways to achieve different compositions. This is particularly useful for students, professionals, and anyone interested in applying combinatorial mathematics to real-world problems.

Calculating Combinations

The formula for calculating combinations is given by:

Cnk n!k!n?k!

Where n is the total number of items, and k is the number of items to be chosen.

Selecting a 5-person Committee from a Group of 7 Men and 9 Women

A committee consisting of 5 men and 6 women can be selected from 7 men and 9 women. We need to explore the different possible combinations:

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

For 2 women and 3 men, the number of ways is calculated as:

C_{6}^{2} cdot C_{7}^{3} frac{6!}{2! cdot 4!} cdot frac{7!}{3! cdot 4!} 15 cdot 35 525

C_{6}^{3} cdot C_{7}^{2} frac{6!}{3! cdot 3!} cdot frac{7!}{2! cdot 5!} 20 cdot 21 420

C_{6}^{4} cdot C_{7}^{1} frac{6!}{4! cdot 2!} cdot frac{7!}{1! cdot 6!} 15 cdot 7 105

C_{6}^{5} cdot C_{7}^{0} frac{6!}{5! cdot 1!} cdot frac{7!}{0! cdot 7!} 6 cdot 1 6

Total: 525 420 105 6 1056 ways

Selecting 5 People from 15 Available

Even without specific gender limitations, the total number of ways to select 5 people from 15 is calculated as:

C_{15}^{5} frac{15!}{5! cdot 10!}

Breaking down the factorials:

15! 15 cdot 14 cdot 13 cdot 12 cdot 11 cdot 10!

5! 5 cdot 4 cdot 3 cdot 2 cdot 1

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

Simplifying:

C_{15}^{5} frac{15 cdot 14 cdot 13 cdot 12 cdot 11 cdot 10!}{5! cdot 10!} frac{15 cdot 14 cdot 13 cdot 12 cdot 11}{5!} 3003

Therefore, there are 3003 ways to form a 5-person committee from 15 people.

Cases with Minimum Men Requirements

A specific committee requires a minimum of 4 men and any number of women. We explore the different scenarios:

4 men and 2 women: C_{9}^{4} cdot C_{7}^{2} 126 cdot 21 2646 5 men and 1 woman: C_{9}^{5} cdot C_{7}^{1} 126 cdot 7 882 6 men and 0 women: C_{9}^{6} cdot C_{7}^{0} 84 cdot 1 84

Total ways: 2646 882 84 3612 ways

At Least 2 Women in a 5-person Committee

For a 5-person committee with at least 2 women, we explore the different cases:

2 women and 3 men: C_{8}^{2} cdot C_{5}^{3} 28 cdot 10 280 3 women and 2 men: C_{8}^{3} cdot C_{5}^{2} 56 cdot 10 560 4 women and 1 man: C_{8}^{4} cdot C_{5}^{1} 70 cdot 5 350 5 women and no men: C_{8}^{5} cdot C_{5}^{0} 56 cdot 1 56

Total ways: 280 560 350 56 1246 ways

By applying these mathematical principles, we can efficiently determine the number of ways to form committees under various constraints, ensuring a thorough understanding of combinations in practical scenarios.