Question

Let a eleven cricket players be chosen out of a batch $15$ players .
$\left(i\right)$ Let "k" be the number of ways a particular is always chosen
$\left(ii\right)$ Let "m" be the number of ways a particular player is never chosen.
Find $k-m-1000$ ?

Answer 1

$\left(i\right)$ Since particular player is always chosen it means that $11-1=10$ players are selected out of the remaining $15-1=14$ players.
$\therefore$ Required number of ways = ${}^{14}{C}_{10}={}^{14}{C}_4=\frac{\mathrm{14.13.12.11}}{\mathrm{1.2.3.4}}=1365$
$\left(ii\right)$ Since particular player is never chosen. It means that $11$ players are selected out of the remaining $15-1=14$ players.
$\therefore$ Required number of ways = ${}^{14}{C}_{11}{=}^{14}{C}_3=\frac{\mathrm{14.13.12}}{\mathrm{1.2.3}}=364.$
$\therefore k-m-1000=1$