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 :

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

Answer 1

The correct option is B $25$

$5$ boys and $n$ girls
Total ways of forming team of $3$ members under given condition
$={}^5{C}_1.{}^n{C}_2+{}^5{C}_2.{}^n{C}_1\Rightarrow {}^5{C}_1.{}^n{C}_2+{}^5{C}_2.{}^n{C}_1=1750\Rightarrow \frac{5n(n-1)}{2}+10n=1750\Rightarrow \frac{n(n-1)}{2}+2n=350\Rightarrow n^2+3n=700\Rightarrow n^2+3n-700=0\Rightarrow (n+28)(n-25)=0\Rightarrow n=25$