Question

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:
1. 28
2. 27
3. 25
4. 24

Answer 1

Correct Answer - Option 3 : 25

From question, the number of ways in which a team of 3 contains at least one boy or at least one girl.

Let ‘N’ denote number of ways.

From question,

⇒ N = 1750

Now,

⇒ N = (1B and 2G) + (2B and 1G) = 1750

⇒ N = (⁵C₁ × ⁿC₂) + (⁵C₂ × ⁿC₁) = 1750

\(\Rightarrow N = \left( {5 \times \left( {\frac{{n\left( {n - 1} \right)}}{2}} \right)} \right) + \left( {10 \times n} \right) = 1750\) 

\(\Rightarrow N = \frac{{5{n^2} - 5n + 20n}}{2} = 1750\) 

\(\Rightarrow N = \frac{{5{n^2} + 15n}}{2} = 1750\) 

⇒ 5n² + 15n = 3500

⇒ 5(n² + 3n) = 3500

⇒ n² + 3n = 700

⇒ n² + 3n – 700 = 0

⇒ n² + 28n – 25n – 700 = 0

⇒ n(n + 28) – 25(n + 28) = 0

⇒ (n – 25)(n + 28) = 0

∴ n = 25, -28

Negative value is not possible.

∴ n = 25