Question

Out of $16$ players of a cricket team, $4$ are bowlers and $2$ are wicket-keepers. A team of $11$ players is to be chosen so as to contain at least $3$ bowlers and at least $1$ wicket-keeper. In how many ways can the team be selected?

Answer 1

Let B stand for bowler, W for wicket-keeper and O for others. Thus we have B (4), W (2), O (10). Now we have to keep in mind the restrictions of at least 3 bowlers so that there could be either 3 or 4 bowlers, at least one wicket-keeper, so that there could be either 1 or 2 wicket-keepers.

B(4)	W(2)	O(10)
3	1	7	${}^4{C}_3\cdot {}^2{C}_1\cdot {}^{10}{C}_7=960$
3	2	6	${}^4{C}_3\cdot {}^2{C}_2\cdot {}^{10}{C}_6=840$
4	1	6	${}^4{C}_4\cdot {}^2{C}_1\cdot {}^{10}{C}_6=420$
4	2	5	${}^4{C}_4\cdot {}^2{C}_2\cdot {}^{10}{C}_5=252$

$\therefore$ Total selection on adding = 2472