Out of 6 boys and 4 girls a group of 7 is to be formed , In how many ways can this be done if the group is to have a majority of boys?

Options

A. 90
B. 100
C. 120
D. 180

Answer

To solve the problem of forming a group of 7 from 6 boys and 4 girls with a majority of boys, we will break down the solution into cases based on the number of boys in the group. A majority of boys means that there must be more boys than girls in the group.

### Step-by-Step Solution:

1. **Identify the Cases**:
Since we need a majority of boys in a group of 7, the possible distributions of boys (B) and girls (G) in the group can be:
- Case 1: 6 boys and 1 girl (6B, 1G)
- Case 2: 5 boys and 2 girls (5B, 2G)
- Case 3: 4 boys and 3 girls (4B, 3G)

(Note: 3 boys and 4 girls would not satisfy the majority condition.)

2. **Case 1: 6 Boys and 1 Girl**:
- Choose 6 boys from 6: \( \binom{6}{6} = 1 \)
- Choose 1 girl from 4: \( \binom{4}{1} = 4 \)
- Total ways for this case:
\[ 1 \times 4 = 4 \]

3. **Case 2: 5 Boys and 2 Girls**:
- Choose 5 boys from 6: \( \binom{6}{5} = 6 \)
- Choose 2 girls from 4: \( \binom{4}{2} = 6 \)
- Total ways for this case:
\[ 6 \times 6 = 36 \]

4. **Case 3: 4 Boys and 3 Girls**:
- Choose 4 boys from 6: \( \binom{6}{4} = 15 \)
- Choose 3 girls from 4: \( \binom{4}{3} = 4 \)
- Total ways for this case:
\[ 15 \times 4 = 60 \]

5. **Total Ways**:
Now, we sum the total ways from all cases:
\[ 4 + 36 + 60 = 100 \]

### Final Answer:
The total number of ways to form a group of 7 with a majority of boys is **100**.

---