1. Form the counting expression

We have a group of $5$ boys and $n$ girls. A team of $3$ students must contain:

• at least one boy, and
• at least one girl.

So the possible compositions of the team are:

• $1$ boy and $2$ girls
• $2$ boys and $1$ girl

Thus, total valid teams:

$\binom{5}{1}\binom{n}{2} + \binom{5}{2}\binom{n}{1} = 1750$

2. Substitute combination values

$\binom{5}{1} = 5, \quad \binom{5}{2} = 10$

So,

$5\binom{n}{2} + 10n = 1750$

Using

$\binom{n}{2} = \frac{n(n-1)}{2}$

we get

$5\cdot \frac{n(n-1)}{2} + 10n = 1750$

3. Simplify the equation

$\frac{5}{2}n(n-1) + 10n = 1750$

Multiply throughout by $2$:

$5n(n-1) + 20n = 3500$

$5n^2 - 5n + 20n = 3500$

$5n^2 + 15n = 3500$

Divide by $5$:

$n^2 + 3n = 700$

$n^2 + 3n - 700 = 0$

4. Solve the quadratic

Factorize:

$n^2 + 28n - 25n - 700 = 0$

$n(n+28) - 25(n+28) = 0$

$(n-25)(n+28) = 0$

So,

$n = 25 \quad \text{or} \quad n = -28$

Since number of girls cannot be negative,

$n = 25$

5. Check with options

Option B is $25$.

6. Comparison with stored answer

Stored correct answer: B

Our derived answer is also B.