Question

There are $15$ players in a cricket team, out of which $6$ are bowlers, $7$ are batsmen and $2$ are wicket keepers. The number of ways, a team of $11$ players be selected from them so as to include at least $4$ bowlers, $5$ batsmen and $1$ wicketkeeper, is _______.

Answer 1

$15:$ Players
$6:$ Bowlers
$7:$ Batsman
$2:$ Wicket keepers
Total number of ways for:
at least $4$ bowlers, $5$ batsman $\& 1$ wicket keeper
$={ }^{6} C_{4}\left({ }^{7} C_{6} \times{ }^{2} C_{1}+{ }^{7} C_{5} \times{ }^{2} C_{2}\right)+{ }^{6} C_{5} \times{ }^{7} C_{5} \times{ }^{2} C_{1} $
$=777$