From 7 men and 4 women a committee of 6 is to be formed such that the committee contains at least two women.What is the number of ways to do this?

Options

A. 210
B. 371
C. 462
D. 5544

Answer

To solve the problem of forming a committee of 6 members from 7 men and 4 women with the condition that there are at least 2 women, we can break it down into different cases based on the number of women in the committee.

### Step-by-Step Solution:

1. **Identify the Cases**:
We need to consider the following cases for the committee composition:
- Case 1: 2 women and 4 men
- Case 2: 3 women and 3 men
- Case 3: 4 women and 2 men

2. **Calculate for Case 1 (2 Women and 4 Men)**:
- The number of ways to choose 2 women from 4:
\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]
- The number of ways to choose 4 men from 7:
\[ \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]
- Total ways for Case 1:
\[ \text{Total for Case 1} = \binom{4}{2} \times \binom{7}{4} = 6 \times 35 = 210 \]

3. **Calculate for Case 2 (3 Women and 3 Men)**:
- The number of ways to choose 3 women from 4:
\[ \binom{4}{3} = \frac{4!}{3!(4-3)!} = 4 \]
- The number of ways to choose 3 men from 7:
\[ \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]
- Total ways for Case 2:
\[ \text{Total for Case 2} = \binom{4}{3} \times \binom{7}{3} = 4 \times 35 = 140 \]

4. **Calculate for Case 3 (4 Women and 2 Men)**:
- The number of ways to choose 4 women from 4:
\[ \binom{4}{4} = 1 \]
- The number of ways to choose 2 men from 7:
\[ \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21 \]
- Total ways for Case 3:
\[ \text{Total for Case 3} = \binom{4}{4} \times \binom{7}{2} = 1 \times 21 = 21 \]

5. **Combine All Cases**:
- Total number of ways to form the committee with at least 2 women:
\[ \text{Total} = \text{Total for Case 1} + \text{Total for Case 2} + \text{Total for Case 3} \]
\[ = 210 + 140 + 21 = 371 \]

### Final Answer:
The total number of ways to form the committee is **371**.