Question

In how many ways can one select a cricket team of eleven from $17$ players in which only $5$ players can bowl if each cricket team of $11$ must include exactly $4$ bowlers?

Answer 1

Given $17$ players , $5$ players are bowlers , so remaining $12$ are batsman.

A cricket team of $11$ players is to be selected in such a way that there are exactly $4$ bowlers.

$4$ bowlers can be selected in ${}^5{C}_4$ways and the remaining $7$ players can be selected out of the $12$ players in ${}^{12}{C}_7$ ways.

Thus, by multiplication principle, required number of ways of selecting cricket team

$\Rightarrow {}^5{C}_4{\times }^{12}{C}_7=\frac{5!}{4!1!}\times \frac{12!}{7!5!}$
$=\frac{12\times 11\times 10\times 9\times 8}{5\times 4\times 3\times 2\times 1}\times 1$

$=3960$