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
exactly 3 girls

Answer

To solve the problem of forming a committee of 7 members consisting of exactly 3 girls from a group of 9 boys and 4 girls, we can follow these steps:

### Step 1: Select 3 Girls from 4
We need to choose exactly 3 girls from the available 4 girls. The number of ways to choose 3 girls from 4 is given by the combination formula:

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

Calculating this:

\[ \binom{4}{3} = \frac{4!}{3!(4-3)!} = \frac{4!}{3! \cdot 1!} = \frac{4 \times 3!}{3! \times 1} = 4 \]

### Step 2: Select 4 Boys from 9
After selecting 3 girls, we need to select 4 boys from the 9 available boys. The number of ways to choose 4 boys from 9 is given by:

\[ \text{Number of ways to choose 4 boys} = \binom{9}{4} \]

Calculating this:

\[ \binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9!}{4! \cdot 5!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = \frac{3024}{24} = 126 \]

### Step 3: Calculate Total Combinations
Now, we multiply the number of ways to choose the girls by the number of ways to choose the boys to get the total number of ways to form the committee:

\[ \text{Total ways} = \binom{4}{3} \times \binom{9}{4} = 4 \times 126 = 504 \]

### Final Answer
Thus, the total number of ways to form the committee of 7 members with exactly 3 girls is **504**.

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