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

Out of 17 players, 5 players are bowlers.
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 $^5C_4$ ways and the remaining 7 players can be selected out of the 12 players in $^{12}C_7$
Thus, by multiplication principle, required number of ways of selecting cricket team
$=$ $^5C_4\times\space^{12}C_7$

$=\frac{5!}{4!1!}\times\frac{12!}{7!5!}$

$=\frac{5\times12\times11\times10\times9\times8}{5\times4\times3\times2\times1}$
$=3960$