The number of ways in which a cricket team of | vedclass

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Question

(D) We need to select $11$ members from $6$ batsmen $(B)$,$6$ bowlers $(Bo)$,$4$ all-rounders $(A)$,and $4$ wicket keepers $(W)$.Constraints: $B \ge 4

Vedclass · Accepted Answer

$13080$

Vedclass · Answer

(D) We need to select $11$ members from $6$ batsmen $(B)$,$6$ bowlers $(Bo)$,$4$ all-rounders $(A)$,and $4$ wicket keepers $(W)$.Constraints: $B \ge 4$,$Bo \ge 3$,$A \ge 2$,$W = 1$.Total members selected = $11$.Let $b, bo, a, w$ be the number of players selected from each category.$w = 1$,so $b + bo + a = 10$.Possible cases $(b, bo, a)$ such that $b \ge 4, bo \ge 3, a \ge 2$:Case $1$: $(5, 3, 2) \implies \binom{6}{5} 	imes \binom{6}{3} 	imes \binom{4}{2} 	imes \binom{4}{1} = 6 	imes 20 	imes 6 	imes 4 = 2880$.Case $2$: $(4, 4, 2) \implies \binom{6}{4} 	imes \binom{6}{4} 	imes \binom{4}{2} 	imes \binom{4}{1} = 15 	imes 15 	imes 6 	imes 4 = 5400$.Case $3$: $(4, 3, 3) \implies \binom{6}{4} 	imes \binom{6}{3} 	imes \binom{4}{3} 	imes \binom{4}{1} = 15 	imes 20 	imes 4 	imes 4 = 4800$.Total ways = $2880 + 5400 + 4800 = 13080$.

The number of ways in which a cricket team of $11$ members can be formed out of $6$ batsmen,$6$ bowlers,$4$ all-rounders and $4$ wicket keepers by selecting at least $4$ batsmen,at least $3$ bowlers,at least $2$ all-rounders and only $1$ wicket keeper is:

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