In how many ways can a committee of 6 members be formed from 7 men and 6 ladies consisting of 4 men and 2 ladies ?

Options

A. a)252
B. b)525
C. c)625
D. d)256

Answer

To solve the problem of forming a committee of 6 members consisting of 4 men and 2 ladies from a group of 7 men and 6 ladies, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Groups**: We have 7 men and 6 ladies. We need to select 4 men and 2 ladies to form a committee of 6 members.

2. **Calculate the Number of Ways to Choose Men**:
To select 4 men from 7 men, we use the combination formula:
\[ \text{Number of ways to choose 4 men} = \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7!}{4! \cdot 3!} \]
Calculating this gives:
\[ \binom{7}{4} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

3. **Calculate the Number of Ways to Choose Ladies**:
To select 2 ladies from 6 ladies, we again use the combination formula:
\[ \text{Number of ways to choose 2 ladies} = \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6!}{2! \cdot 4!} \]
Calculating this gives:
\[ \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15 \]

4. **Combine the Selections**:
The total number of ways to form the committee is the product of the number of ways to choose the men and the number of ways to choose the ladies:
\[ \text{Total ways} = \binom{7}{4} \times \binom{6}{2} = 35 \times 15 \]

5. **Calculate the Final Result**:
\[ 35 \times 15 = 525 \]

Thus, the total number of ways to form the committee is **525**.