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 the condition that women must be in the majority, we can break it down into cases based on the number of women selected.

### Step-by-Step Solution:

1. **Identify Cases for Women in Majority**:
- Women must be more than half of the committee. Since the committee has 12 members, women must be at least 7. The possible distributions are:
- Case 1: 9 women and 3 men
- Case 2: 8 women and 4 men
- Case 3: 7 women and 5 men

2. **Calculate Case 1: 9 Women and 3 Men**:
- The number of ways to choose 9 women from 9 is \( \binom{9}{9} = 1 \).
- The number of ways to choose 3 men from 8 is \( \binom{8}{3} \).
- Calculate \( \binom{8}{3} \):
\[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]
- Total ways for Case 1: \( 1 \times 56 = 56 \).

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

4. **Calculate Case 3: 7 Women and 5 Men**:
- The number of ways to choose 7 women from 9 is \( \binom{9}{7} = 36 \).
- The number of ways to choose 5 men from 8 is \( \binom{8}{5} = \binom{8}{3} = 56 \) (since \( \binom{n}{r} = \binom{n}{n-r} \)).
- Total ways for Case 3: \( 36 \times 56 = 2016 \).

5. **Total Ways to Form the Committee**:
- Add the number of ways from all cases:
\[ \text{Total} = 56 + 630 + 2016 = 2702 \]

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

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