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 from 9 boys and 4 girls, with the condition that the committee must consist of exactly 3 girls, we can follow these steps:

### Step 1: Determine the number of girls to select
We need to select exactly 3 girls from the 4 available girls. This can be represented mathematically as \( \binom{4}{3} \).

### Step 2: Calculate the number of ways to choose the girls
Using the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \):
\[ \binom{4}{3} = \frac{4!}{3!(4-3)!} = \frac{4!}{3! \cdot 1!} = \frac{4 \times 3!}{3! \times 1} = 4 \]

### Step 3: Determine the number of boys to select
After selecting 3 girls, we need to select the remaining members of the committee from boys. Since the committee consists of 7 members in total, we need to select \( 7 - 3 = 4 \) boys from the 9 available boys. This can be represented as \( \binom{9}{4} \).

### Step 4: Calculate the number of ways to choose the boys
Using the combination formula again:
\[ \binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9!}{4! \cdot 5!} \]
Calculating \( \binom{9}{4} \):
\[ = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = \frac{3024}{24} = 126 \]

### Step 5: Calculate the total number of ways to form the committee
Now, we multiply the number of ways to choose the girls by the number of ways to choose the boys:
\[ \text{Total ways} = \binom{4}{3} \times \binom{9}{4} = 4 \times 126 = 504 \]

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

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