Forming a 5-Man Committee: A Comprehensive Guide to Combinatorial Mathematics

Introduction

When forming a committee from a larger group of individuals, the order in which people are chosen does not matter. This problem falls under the realm of combinatorial mathematics, specifically combinations. In this article, we will explore how to determine the number of ways to form a 5-man committee from a group of 8 people using the concept of combinations.

The Combinatorial Formula

The number of ways to choose r elements from n elements is given by the combination formula:

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

In the context of our problem, n 8 and r 5. Therefore, we need to calculate (binom{8}{5}) as follows:

(binom{8}{5} frac{8!}{5!8-5!} frac{8!}{5!3!})

Factorial Calculations

Next, let's calculate the factorials:

(8! 8 times 7 times 6 times 5!)

(5! 5 times 4 times 3 times 2 times 1 120)

(3! 3 times 2 times 1 6)

Substituting these values into the combination formula, we get:

(binom{8}{5} frac{8 times 7 times 6}{3 times 2 times 1} frac{336}{6} 56))

Thus, there are 56 ways to form a 5-man committee from 8 people.

Enumerating Possibilities

To further illustrate, let's enumerate all possible 5-man committees from a group of 8 people: A, B, C, D, E, F, G, H. Here are the combinations:

A, B, C, D, E A, B, C, D, F A, B, C, D, G A, B, C, D, H A, B, C, E, F A, B, C, E, G A, B, C, E, H A, B, C, F, G A, B, C, F, H A, B, C, G, H A, B, D, E, F A, B, D, E, G A, B, D, E, H A, B, D, F, G A, B, D, F, H A, B, D, G, H A, B, E, F, G A, B, E, F, H A, B, E, G, H A, B, F, G, H A, C, D, E, F A, C, D, E, G A, C, D, E, H A, C, D, F, G A, C, D, F, H A, C, D, G, H A, C, E, F, G A, C, E, F, H A, C, E, G, H A, C, F, G, H A, D, E, F, G A, D, E, F, H A, D, E, G, H A, D, F, G, H A, E, F, G, H B, C, D, E, F B, C, D, E, G B, C, D, E, H B, C, D, F, G B, C, D, F, H B, C, D, G, H B, C, E, F, G B, C, E, F, H B, C, E, G, H B, C, F, G, H B, D, E, F, G B, D, E, F, H B, D, E, G, H B, D, F, G, H B, E, F, G, H C, D, E, F, G C, D, E, F, H C, D, E, G, H C, D, F, G, H D, E, F, G, H

As seen above, there are 56 unique combinations.

Pitfalls in Using Order

Earlier, a contributor suggested a different approach where the order of picking people matters, resulting in the calculation 8x7x6x5x4 - 6720 ways. This method is incorrect for forming committees, as order does not matter. Let's understand why:

When order matters, this problem is a permutation problem, where each of the 5 people is distinct. In such cases, the number of ways to form a 5-man committee is 8P5, which is 8! / 3!. However, since we are forming a committee where order does not matter, this problem is a combination problem, as shown in the correct calculation above.

To summarize, the correct number of ways to form a 5-man committee from 8 people is 56, derived using the combination formula and by understanding the problem context accurately.