A committee consisting of 3 men and 5 women is to be selected from 5 men and 10 women. Find how many ways the committee can be formed.

1. Problem Description

A committee consisting of 3 men and 5 women is to be selected from 5 men and 10 women. Find how many ways the committee can be formed.

2. Solution Steps

We need to choose 3 men from 5 men, and 5 women from 10 women. The number of ways to choose $k$ objects from a set of $n$ objects is given by the combination formula:

$C(n, k) = \frac{n!}{k!(n-k)!}$

First, we calculate the number of ways to choose 3 men from 5 men:

$C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = \frac{5 \times 4 \times 3!}{3! \times 2 \times 1} = \frac{5 \times 4}{2} = 10$

Next, we calculate the number of ways to choose 5 women from 10 women:

$C(10, 5) = \frac{10!}{5!(10-5)!} = \frac{10!}{5!5!} = \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5!}{5! \times 5 \times 4 \times 3 \times 2 \times 1} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = \frac{10 \times 9 \times 8 \times 7 \times 6}{120} = 2 \times 9 \times 2 \times 7 = 252$

Since we choose the men and the women independently, we multiply the number of ways to choose the men and the number of ways to choose the women to get the total number of ways to form the committee.

Total number of ways = $C(5, 3) \times C(10, 5) = 10 \times 252 = 2520$

3. Final Answer

The committee can be formed in 2520 ways.