Question

Out of $n$ students, a committee of $12$ students is formed. If number of such committees containing $2$ particular students $A,B$ is $3$ times the number of committees containing another $3$ particular students $D,E,F,$ then $n$ is

• A. $32$
• B. $38$
• C. $45$
• D. $72$

Answer 1

The correct option is A $32$

Out of $n$ students, $A,B$ are always included.
Other $10$ students from $(n-2)$ students can be selected in ${}^{n-2}{C}_{10}$ ways.

Similarly, the other committee can be formed in ${}^{n-3}{C}_9$ ways.
Given that ${}^{n-2}{C}_{10}=3\times {}^{n-3}{C}_9$
$\Rightarrow \frac{(n-2)!}{(n-12)!10!}=3\times \frac{(n-3)!}{(n-12)!9!}$
$\Rightarrow \frac{n-2}{10}=3$
$\Rightarrow n=32$