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Q.

Among 22 cricket players, there are 3 wicket keepers and 6 bowlers. In how many ways can a team of 11 players be chosen so as to include exactly one wicketkeeper and at least 4 bowlers?

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a

2145

b

23166

c

77220

d

102531

answer is D.

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Detailed Solution

Total number of players = 22
Number of wicket keepers = 3
Number of bowlers = 6
Total number of players required for cricket team = 11
13 are neither bowler nor wicketkeeper.
we require at least 4 bowlers.
Case (i)
In this case we choose 4 bowlers, 1 wicket keeper and 6 batsman out of 6 bowlers, 3 wicket keepers and 13 batsman as

C_{4}^{6} \times C_{1}^{3} \times C_{6}^{13} = 77220

.
Case (ii)
In this case we choose 5 bowlers, 1 wicket keeper and 5 batsman out of 6 bowlers, 3 wicket keepers and 13 batsman as

C_{5}^{6} \times C_{1}^{3} \times C_{5}^{13} = 23166

.
Case (iii)
In this case we choose 6 bowlers, 1 wicket keeper and 4 batsman out of 6 bowlers, 3 wicket keepers and 13 batsman as

C_{6}^{6} \times C_{1}^{3} \times C_{4}^{13} = 2145

.
Total ways = 77220+23166+2145 = 102531

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