In how many different ways can a committee of 8 persons be formed out of 5 men and 3 women?

1. Break Down the Problem

We need to determine the number of ways to form a committee of 8 persons using a mix of 5 men and 3 women. Since the total number of individuals (8) is greater than those available (5 men + 3 women = 8), we can only form a committee with all available persons.

2. Relevant Concepts

The concept we will utilize here is combinations, specifically the binomial coefficient, which is defined as: $\binom{n}{r} = \frac{n!}{r! \cdot (n - r)!}$ where $n$ is the total number of items to choose from, and $r$ is the number of items to choose.

3. Analysis and Detail

In this scenario:

• We have $n = 8$ (the total number of individuals: 5 men + 3 women).
• We want to choose $r = 8$.

The binomial coefficient can be calculated as: $\binom{8}{8} = \frac{8!}{8! \cdot (8 - 8)!} = \frac{8!}{8! \cdot 0!} = \frac{1}{1} = 1$

4. Verify and Summarize

There is exactly one way to choose all 8 persons from a set of 8, which is indeed the only option available.

Final Answer

There is 1 different way to form a committee of 8 persons out of 5 men and 3 women.