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 atmost three girls

Answer

To solve the problem of forming a committee of 7 members from 9 boys and 4 girls with at most 3 girls, we can break it down into cases based on the number of girls in the committee. The cases are:

1. **Case 1**: 0 girls and 7 boys
2. **Case 2**: 1 girl and 6 boys
3. **Case 3**: 2 girls and 5 boys
4. **Case 4**: 3 girls and 4 boys

We will calculate the number of ways for each case and then sum them up.

### Step 1: Calculate the number of ways for each case

**Case 1**: 0 girls and 7 boys
- The number of ways to choose 0 girls from 4 is given by \( \binom{4}{0} \).
- The number of ways to choose 7 boys from 9 is given by \( \binom{9}{7} \).

\[ \text{Ways for Case 1} = \binom{4}{0} \times \binom{9}{7} = 1 \times 36 = 36 \]

**Case 2**: 1 girl and 6 boys
- The number of ways to choose 1 girl from 4 is given by \( \binom{4}{1} \).
- The number of ways to choose 6 boys from 9 is given by \( \binom{9}{6} \).

\[ \text{Ways for Case 2} = \binom{4}{1} \times \binom{9}{6} = 4 \times 84 = 336 \]

**Case 3**: 2 girls and 5 boys
- The number of ways to choose 2 girls from 4 is given by \( \binom{4}{2} \).
- The number of ways to choose 5 boys from 9 is given by \( \binom{9}{5} \).

\[ \text{Ways for Case 3} = \binom{4}{2} \times \binom{9}{5} = 6 \times 126 = 756 \]

**Case 4**: 3 girls and 4 boys
- The number of ways to choose 3 girls from 4 is given by \( \binom{4}{3} \).
- The number of ways to choose 4 boys from 9 is given by \( \binom{9}{4} \).

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

### Step 2: Sum the number of ways from all cases

Now, we sum the number of ways from all four cases:

\[ \text{Total Ways} = \text{Ways for Case 1} + \text{Ways for Case 2} + \text{Ways for Case 3} + \text{Ways for Case 4} \]

\[ \text{Total Ways} = 36 + 336 + 756 + 504 = 1632 \]

### Final Answer

The total number of ways to form the committee is **1632**.

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