The number of committees of 5 persons consisting of at least one female number, that can be formed from 6 males and 4 females, is

Options

A. 246
B. 252
C. 6
D. none of these

Answer

To solve the problem of forming committees of 5 persons consisting of at least one female from a group of 6 males and 4 females, we can break down the solution into several cases based on the number of females in the committee.

### Step-by-Step Solution:

1. **Identify the Total Members**: We have 6 males (M) and 4 females (F). We need to form a committee of 5 members with at least one female.

2. **Consider Cases Based on the Number of Females**:
- **Case 1**: 1 female and 4 males
- **Case 2**: 2 females and 3 males
- **Case 3**: 3 females and 2 males
- **Case 4**: 4 females and 1 male

3. **Calculate Each Case**:
- **Case 1**:
- Choose 1 female from 4: \( \binom{4}{1} \)
- Choose 4 males from 6: \( \binom{6}{4} \)
- Total for Case 1:
\[ \binom{4}{1} \times \binom{6}{4} = 4 \times 15 = 60 \]

- **Case 2**:
- Choose 2 females from 4: \( \binom{4}{2} \)
- Choose 3 males from 6: \( \binom{6}{3} \)
- Total for Case 2:
\[ \binom{4}{2} \times \binom{6}{3} = 6 \times 20 = 120 \]

- **Case 3**:
- Choose 3 females from 4: \( \binom{4}{3} \)
- Choose 2 males from 6: \( \binom{6}{2} \)
- Total for Case 3:
\[ \binom{4}{3} \times \binom{6}{2} = 4 \times 15 = 60 \]

- **Case 4**:
- Choose 4 females from 4: \( \binom{4}{4} \)
- Choose 1 male from 6: \( \binom{6}{1} \)
- Total for Case 4:
\[ \binom{4}{4} \times \binom{6}{1} = 1 \times 6 = 6 \]

4. **Add All Cases Together**:
- Total number of committees:
\[ 60 + 120 + 60 + 6 = 246 \]

### Final Answer:
The total number of committees of 5 persons consisting of at least one female is **246**.

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