Three men (out of 7) and 3 women (out of 6) will be chosen to serve on a committee. In how many ways can the committee be formed?

Answer

To solve the problem of forming a committee of 3 men from a group of 7 men and 3 women from a group of 6 women, we will use the concept of combinations.

### Step-by-Step Solution:

1. **Identify the Groups**:
- We have 7 men and we need to choose 3.
- We have 6 women and we need to choose 3.

2. **Use the Combination Formula**:
The combination formula is given by:
\[ nCr = \frac{n!}{(n - r)! \cdot r!} \]
where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

3. **Calculate the Number of Ways to Choose Men**:
- For the men, we have \( n = 7 \) and \( r = 3 \).
\[ \text{Number of ways to choose 3 men from 7} = 7C3 = \frac{7!}{(7 - 3)! \cdot 3!} = \frac{7!}{4! \cdot 3!} \]
- Simplifying this:
\[ 7C3 = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = \frac{210}{6} = 35 \]

4. **Calculate the Number of Ways to Choose Women**:
- For the women, we have \( n = 6 \) and \( r = 3 \).
\[ \text{Number of ways to choose 3 women from 6} = 6C3 = \frac{6!}{(6 - 3)! \cdot 3!} = \frac{6!}{3! \cdot 3!} \]
- Simplifying this:
\[ 6C3 = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = \frac{120}{6} = 20 \]

5. **Calculate the Total Number of Ways to Form the Committee**:
- Since the selections of men and women are independent, we multiply the number of ways to choose men by the number of ways to choose women:
\[ \text{Total ways} = (7C3) \times (6C3) = 35 \times 20 = 700 \]

### Final Answer:
The total number of ways to form the committee is **700**.