There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsmen and 2 are wicketkeepers.…

We have,

$15 :$ Players

$6 :$ Bowlers

$7 :$ Batsman

$2 :$ Wicket keepers

Total number of ways for at least $4$ bowlers, $5$ batsman $$and $1$ wicket keeper

$=\left({}^6\mathrm{C}_4\times {}^7\mathrm{C}_6\times {}^2\mathrm{C}_1\right)+\left({}^6\mathrm{C}_4\times {}^7\mathrm{C}_5\times {}^2\mathrm{C}_2\right)+\left({}^6\mathrm{C}_5\times {}^7\mathrm{C}_5\times {}^2\mathrm{C}_1\right)$

$=777$