A committee of 7 members has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of
at least 3 girls

Answer

To solve the problem of forming a committee of 7 members from 9 boys and 4 girls with the condition that there are at least 3 girls, we can break it down into two cases:

### Step 1: Identify the Cases
We need to consider the following two cases:
1. Case 1: 3 girls and 4 boys
2. Case 2: 4 girls and 3 boys

### Step 2: Calculate for Case 1 (3 girls and 4 boys)
- **Selecting 3 girls from 4**: This can be done using the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

\[ \text{Number of ways to choose 3 girls from 4} = \binom{4}{3} = 4 \]

- **Selecting 4 boys from 9**:

\[ \text{Number of ways to choose 4 boys from 9} = \binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126 \]

- **Total ways for Case 1**:

\[ \text{Total for Case 1} = \binom{4}{3} \times \binom{9}{4} = 4 \times 126 = 504 \]

### Step 3: Calculate for Case 2 (4 girls and 3 boys)
- **Selecting 4 girls from 4**:

\[ \text{Number of ways to choose 4 girls from 4} = \binom{4}{4} = 1 \]

- **Selecting 3 boys from 9**:

\[ \text{Number of ways to choose 3 boys from 9} = \binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]

- **Total ways for Case 2**:

\[ \text{Total for Case 2} = \binom{4}{4} \times \binom{9}{3} = 1 \times 84 = 84 \]

### Step 4: Combine the Results
Now, we add the total ways from both cases to find the total number of ways to form the committee:

\[ \text{Total ways} = \text{Total for Case 1} + \text{Total for Case 2} = 504 + 84 = 588 \]

### Final Answer
Thus, the total number of ways to form the committee is **588**.

---