A group consists of 1 girls and 7 boys. In how many ways can a team of 5 members be selected if a team has
(i) no girl

Answer

To solve the problem of selecting a team of 5 members from a group consisting of 1 girl and 7 boys, where the team must have no girls, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Group Composition**:
- We have 1 girl and 7 boys in the group.

2. **Determine Team Composition**:
- Since the team must have no girls, we need to select all 5 members from the boys. This means we need to select 5 boys from the 7 available boys.

3. **Use the Combination Formula**:
- The number of ways to choose \( r \) members from \( n \) members is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n - r)!} \]
- Here, \( n = 7 \) (the number of boys) and \( r = 5 \) (the number of boys we want to select).

4. **Calculate the Combination**:
- Plugging the values into the formula:
\[ \binom{7}{5} = \frac{7!}{5!(7 - 5)!} = \frac{7!}{5! \cdot 2!} \]
- This simplifies to:
\[ \binom{7}{5} = \frac{7 \times 6 \times 5!}{5! \times 2!} = \frac{7 \times 6}{2!} \]
- Since \( 2! = 2 \):
\[ \binom{7}{5} = \frac{7 \times 6}{2} = \frac{42}{2} = 21 \]

5. **Conclusion**:
- Therefore, the number of ways to select a team of 5 members with no girls is **21**.