A committee of 12 members is to be formed from 9 women and 8 men. The number of ways of forming the committee with women in majority is

Options

A. 1008
B. 2702
C. 6062
D. 2352

Answer

To solve the problem of forming a committee of 12 members from 9 women and 8 men with women in the majority, we need to consider the possible combinations of women and men that satisfy the majority condition.

### Step-by-Step Solution:

1. **Identify Majority Condition**:
- For women to be in the majority in a committee of 12 members, there must be more women than men. This means the possible distributions of women and men can be:
- 9 women and 3 men
- 8 women and 4 men
- 7 women and 5 men

2. **Calculate Combinations for Each Case**:
- **Case 1**: 9 women and 3 men
- The number of ways to choose 9 women from 9 is given by \( \binom{9}{9} = 1 \).
- The number of ways to choose 3 men from 8 is given by \( \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \).
- Total for this case = \( 1 \times 56 = 56 \).

- **Case 2**: 8 women and 4 men
- The number of ways to choose 8 women from 9 is given by \( \binom{9}{8} = 9 \).
- The number of ways to choose 4 men from 8 is given by \( \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \).
- Total for this case = \( 9 \times 70 = 630 \).

- **Case 3**: 7 women and 5 men
- The number of ways to choose 7 women from 9 is given by \( \binom{9}{7} = \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \).
- The number of ways to choose 5 men from 8 is given by \( \binom{8}{5} = \binom{8}{3} = 56 \).
- Total for this case = \( 36 \times 56 = 2016 \).

3. **Add Up All Cases**:
- Total number of ways to form the committee with women in majority =
- Case 1: 56
- Case 2: 630
- Case 3: 2016
- Therefore, total = \( 56 + 630 + 2016 = 2702 \).

### Final Answer:
The total number of ways of forming the committee with women in majority is **2702**.