There are 15 players in a cricket team, out of which 6 are bowlers,...

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

$= \left(C4 6 \times C6 7 \times C1 2\right) + \left(C4 6 \times C5 7 \times C2 2\right) + \left(C5 6 \times C5 7 \times C1 2\right)$

$= 777$