Question

A guard of $12$ men is formed from a group of $n$ soldiers in all possible ways. If the number of times two particular soldiers $A$ and $B$ are together on guard is thrice the number of times three particular soldiers $C,D,E$ are together on guard, then $n$ is

Answer 1

Number of ways $A$ and $B$ are together is ${}^{n-2}{C}_{10}$
Number of ways $C,D,E$ are together is ${}^{n-3}{C}_9$

Now, ${}^{n-2}{C}_{10}=3\times {}^{n-3}{C}_9$
$\Rightarrow \frac{(n-2)!}{(n-12)!\times 10!}=3\times \frac{(n-3)!}{(n-12)!\times 9!}\Rightarrow \frac{(n-2)!}{(n-3)!}=3\times 10\Rightarrow n-2=30\Rightarrow n=32$