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 1: Identify the groups
We have:
- 6 men
- 8 women

We need to select:
- 4 men
- 3 women

### Step 2: Use combinations to calculate selections
To find the number of ways to choose the men, we use the combination formula \( nCr \), which is given by:

\[ nCr = \frac{n!}{r!(n-r)!} \]

For the men:
\[ \text{Number of ways to choose 4 men from 6} = 6C4 \]

For the women:
\[ \text{Number of ways to choose 3 women from 8} = 8C3 \]

### Step 3: Calculate the combinations
Now we calculate each combination:

1. **Calculating \( 6C4 \)**:
\[ 6C4 = \frac{6!}{4!(6-4)!} = \frac{6!}{4! \cdot 2!} = \frac{6 \times 5}{2 \times 1} = 15 \]

2. **Calculating \( 8C3 \)**:
\[ 8C3 = \frac{8!}{3!(8-3)!} = \frac{8!}{3! \cdot 5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]

### Step 4: Calculate the total number of ways
Now, we multiply the number of ways to choose the men and the women:

\[ \text{Total ways} = 6C4 \times 8C3 = 15 \times 56 \]

Calculating this gives:

\[ 15 \times 56 = 840 \]

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

---