(a) A cricket team of 11 players is to be formed from 16 players including 4 bowlers and 2 wicket-keepers. In

Question 30 of 48

Q. (a) A cricket team of 11 players is to be formed from 16 players including 4 bowlers and 2 wicket-keepers. In how many different ways can a team be formed so that the team contains atleast 3 bowlers and atleast one wicket-keeper ?

(OR)

(b) X speaks truth 4 out of 5 times. A die is thrown. He reports that there is a six. What is the chance that actually there was a six ?

Tamil Nadu DgeTamil Nadu HSC First Year (DGE) Commerce Board 2022Subjective· 5mImportance★★★★★

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(a) Sum the case counts for $\ge3$ bowlers and $\ge1$ wicket-keeper: $2472$ ways. (b) Bayes' theorem gives $\dfrac{4}{9}$ .

Part (a) — Cricket team selection.

The 16 players are 4 bowlers ( $B$ ), 2 wicket-keepers ( $W$ ) and $16-4-2=10$ others ( $O$ ). We need 11 players with at least 3 bowlers and at least 1 wicket-keeper. Enumerate the allowed combinations:

Bowlers	W-keepers	Others	Count
3 ( $\binom{4}{3}=4$ )	1 ( $\binom{2}{1}=2$ )	7 ( $\binom{10}{7}=120$ )	$4\times2\times120=960$
3 ( $4$ )	2 ( $\binom{2}{2}=1$ )	6 ( $\binom{10}{6}=210$ )	$4\times1\times210=840$
4 ( $\binom{4}{4}=1$ )	1 ( $2$ )	6 ( $210$ )	$1\times2\times210=420$
4 ( $1$ )	2 ( $1$ )	5 ( $\binom{10}{5}=252$ )	$1\times1\times252=252$

$\text{Total}=960+840+420+252=2472.$

Part (b) — Truth-telling and Bayes' theorem.

Let $S=$ "a six actually turns up", $T=$ "X reports a six".

$P(S)=\dfrac16,\ P(S')=\dfrac56,\qquad P(\text{X speaks truth})=\dfrac45,\ P(\text{X lies})=\dfrac15.$ …

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Q.(a) A cricket team of 11 players is to be formed from 16 players including 4 bowlers and 2 wicket-keepers. In how many different ways can a team be formed so that the team contains atleast 3 bowlers and atleast one wicket-keeper ?

Concept understanding — Combinations (nCr) and Their Properties

Q.(a) A cricket team of 11 players is to be formed from 16 players including 4 bowlers and 2 wicket-keepers. In how many different ways can a team be formed so that the team contains atleast 3 bowlers and atleast one wicket-keeper ?

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