Question

A guard of 12 men is formed from a group of n soldiers in all possible ways. Find :
(i)The number of times two particular soldiers A and B are together on guard.
(ii)The number of times' three particular soldiers C, D, F are together on guard.
(iii)Also find h if it is found that A and B are three times as often together on guard as C, D,E are.

Answer 1

(i), A and B are together and so out of remaining n -2
we have to choose 10.
${}^{n-2}{C}_{10}$ = $\frac{(n-2)!}{\left(10\right)!(n-12)!}$
(ii) C, D, E are together and so out of remaining n - 3 we have to choose 9
$\therefore$ ${}^{n-3}{C}_9$ = $\frac{(n-3)!}{9!(n-12)!}$
(iii)By given condition $\frac{(n-2)}{\left(10\right)!(n-12)}$ = 3.$\frac{(n-3)}{9!(n-12)!}$
or n - 2 = 30 or n = 32.