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

Answer: 777

$15:$ Players
$6:$ Bowlers
$7:$ Batsman
$2:$ Wicket keepers
Total number of ways for:
at least $4$ bowlers, $5$ batsman $\&1$ wicket keeper
$={}^6{C}_4\left({}^7{C}_6\times {}^2{C}_1+{}^7{C}_5\times {}^2{C}_2\right)+{}^6{C}_5\times {}^7{C}_5\times {}^2{C}_1$
$=777$