In how many ways can a committee of 4 men and 3 women be appointed from 6 men and 8 women ?

Options

A. 480
B. 308
C. 840
D. 640

Answer

To solve the problem of how many ways a committee of 4 men and 3 women can be appointed from 6 men and 8 women, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the total number of men and women**:
- Total men = 6
- Total women = 8

2. **Determine the number of men and women needed for the committee**:
- Men needed = 4
- Women needed = 3

3. **Calculate the number of ways to choose 4 men from 6**:
- The number of ways to choose 4 men from 6 is given by the combination formula \( \binom{n}{r} \):
\[ \binom{6}{4} = \frac{6!}{4!(6-4)!} = \frac{6!}{4! \cdot 2!} \]
- Simplifying this:
\[ = \frac{6 \times 5}{2 \times 1} = 15 \]

4. **Calculate the number of ways to choose 3 women from 8**:
- The number of ways to choose 3 women from 8 is:
\[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3! \cdot 5!} \]
- Simplifying this:
\[ = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]

5. **Combine the two results to find the total number of ways to form the committee**:
- The total number of ways to form the committee is the product of the two combinations:
\[ \text{Total ways} = \binom{6}{4} \times \binom{8}{3} = 15 \times 56 \]

6. **Calculate the final result**:
- Now, calculate \( 15 \times 56 \):
\[ 15 \times 56 = 840 \]

### Final Answer:
The total number of ways to appoint a committee of 4 men and 3 women from 6 men and 8 women is **840**.

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