A committee of 5 is to be formed from a group of 12 students consisting of 8 boys and 4 girls in how many ways can the committee be formed it
contains at least 3 girls ?

Answer

To solve the problem of forming a committee of 5 members from a group of 12 students (8 boys and 4 girls) such that the committee contains at least 3 girls, we can break it down into two cases:

### Step-by-Step Solution:

1. **Identify the Cases**:
- Case 1: The committee has 3 girls and 2 boys.
- Case 2: The committee has 4 girls and 1 boy.

2. **Calculate for Case 1 (3 Girls and 2 Boys)**:
- **Choose 3 girls from 4**:
\[ \text{Number of ways} = \binom{4}{3} = \frac{4!}{3!(4-3)!} = \frac{4!}{3! \cdot 1!} = 4 \]
- **Choose 2 boys from 8**:
\[ \text{Number of ways} = \binom{8}{2} = \frac{8!}{2!(8-2)!} = \frac{8!}{2! \cdot 6!} = \frac{8 \times 7}{2 \times 1} = 28 \]
- **Total ways for Case 1**:
\[ \text{Total for Case 1} = 4 \times 28 = 112 \]

3. **Calculate for Case 2 (4 Girls and 1 Boy)**:
- **Choose 4 girls from 4**:
\[ \text{Number of ways} = \binom{4}{4} = 1 \]
- **Choose 1 boy from 8**:
\[ \text{Number of ways} = \binom{8}{1} = 8 \]
- **Total ways for Case 2**:
\[ \text{Total for Case 2} = 1 \times 8 = 8 \]

4. **Combine the Cases**:
- **Total number of ways to form the committee**:
\[ \text{Total} = \text{Total for Case 1} + \text{Total for Case 2} = 112 + 8 = 120 \]

### Final Answer:
The total number of ways to form the committee containing at least 3 girls is **120**.

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