A group of students comprises of 5 boys and n girls. If the number of ways, in which a team of 3 students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is 1750, then n is equal to:

Options

A. 28
B. 27
C. 25
D. 24

Answer

To solve the problem, we need to find the value of \( n \) (the number of girls) in a group consisting of 5 boys and \( n \) girls, such that the number of ways to select a team of 3 students with at least one boy and one girl is 1750.

### Step-by-Step Solution:

1. **Understanding the Selection Criteria**:
We need to select a team of 3 students that includes at least one boy and one girl. This can happen in two scenarios:
- Selecting 1 boy and 2 girls.
- Selecting 2 boys and 1 girl.

2. **Calculating the Number of Ways for Each Scenario**:
- **Scenario 1**: Selecting 1 boy and 2 girls.
- The number of ways to select 1 boy from 5 boys is \( \binom{5}{1} \).
- The number of ways to select 2 girls from \( n \) girls is \( \binom{n}{2} \).
- Therefore, the total ways for this scenario is:
\[ \binom{5}{1} \cdot \binom{n}{2} = 5 \cdot \frac{n(n-1)}{2} = \frac{5n(n-1)}{2} \]

- **Scenario 2**: Selecting 2 boys and 1 girl.
- The number of ways to select 2 boys from 5 boys is \( \binom{5}{2} \).
- The number of ways to select 1 girl from \( n \) girls is \( \binom{n}{1} \).
- Therefore, the total ways for this scenario is:
\[ \binom{5}{2} \cdot \binom{n}{1} = 10 \cdot n \]

3. **Setting Up the Equation**:
The total number of ways to select the team is the sum of the ways from both scenarios:
\[ \frac{5n(n-1)}{2} + 10n = 1750 \]

4. **Clearing the Fraction**:
To eliminate the fraction, multiply the entire equation by 2:
\[ 5n(n-1) + 20n = 3500 \]

5. **Expanding and Rearranging**:
Expanding the left side gives:
\[ 5n^2 - 5n + 20n = 3500 \]
Combining like terms results in:
\[ 5n^2 + 15n - 3500 = 0 \]

6. **Dividing by 5**:
Simplifying the equation by dividing everything by 5:
\[ n^2 + 3n - 700 = 0 \]

7. **Factoring the Quadratic Equation**:
We can factor the quadratic equation:
\[ (n + 28)(n - 25) = 0 \]
This gives us two potential solutions for \( n \):
\[ n + 28 = 0 \quad \Rightarrow \quad n = -28 \quad (\text{not valid since } n \text{ must be non-negative}) \]
\[ n - 25 = 0 \quad \Rightarrow \quad n = 25 \]

8. **Conclusion**:
Therefore, the number of girls \( n \) is \( 25 \).

### Final Answer:
\( n = 25 \)