In how many ways can a team of 4 people be formed from 4 boys & 5 girls such that girls are never less than boys and team has both boys and girls ?

Options

A. 140
B. 40
C. 60
D. 100

Answer

To solve the problem of forming a team of 4 people from 4 boys and 5 girls, where the number of girls is never less than the number of boys and the team must consist of both boys and girls, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the conditions**:
- The team must consist of both boys and girls.
- The number of girls must be greater than the number of boys.

2. **Determine possible combinations**:
- Since the total number of team members is 4, the possible distributions of boys (B) and girls (G) that satisfy the conditions are:
- 1 boy and 3 girls (1B, 3G)
- 2 boys and 2 girls (2B, 2G)
- The distribution of 3 boys and 1 girl (3B, 1G) is not allowed because it violates the condition that the number of girls must be greater than the number of boys.

3. **Calculate the number of ways for each distribution**:
- **For 1 boy and 3 girls**:
- Number of ways to choose 1 boy from 4 boys:
\[ \binom{4}{1} = 4 \]
- Number of ways to choose 3 girls from 5 girls:
\[ \binom{5}{3} = 10 \]
- Total ways for this case:
\[ 4 \times 10 = 40 \]

- **For 2 boys and 2 girls**:
- Number of ways to choose 2 boys from 4 boys:
\[ \binom{4}{2} = 6 \]
- Number of ways to choose 2 girls from 5 girls:
\[ \binom{5}{2} = 10 \]
- Total ways for this case:
\[ 6 \times 10 = 60 \]

4. **Add the total number of ways from both cases**:
- Total number of ways to form the team:
\[ 40 + 60 = 100 \]

### Final Answer:
The total number of ways to form a team of 4 people such that the number of girls is never less than the number of boys and includes both boys and girls is **100**.