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Question
There are 3 wicketkeepers and 5 bowlers among 22 cricket players. A team of 11 players is to be selected so that there is exactly one wicketkeeper and at least 4 bowlers in the team. How many different teams can be formed?
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Solution
Total number of cricket players = 22
Number of wicketkeepers = 3
Number of bowlers = 5
Remaining players = 14
A team of 11 players consisting of exactly one wicketkeeper and at least 4 bowlers can be selected as follows
(I) 1 wicketkeeper, 4 bowlers, 6 players
or (II) 1 wicketkeeper, 5 bowlers, 5 players
Now, number of selections in (I)
= `""^3"C"_1 xx ""^5 "C"_4 xx ""^14"C"_6`
= `3 xx 5 xx (14 xx 13 xx 12 xx 11 xx 10 xx 9)/(6 xx 5 xx 4 xx 3 xx 2 xx 1)`
= `(15 xx 14 xx 13 xx 2 xx 11 xx 3)/4`
= 45045
Number of selections in (II)
= `""^3"C"_1 xx ""^5 "C"_5 xx ""^14"C"_5`
= `3 xx 1 xx (14 xx 13 xx 12 xx 11 xx 10)/(5 xx 4 xx 3 xx 2 xx 1)`
= 6006
By addition principle, total number of ways choosing a team of 11 players
= 45045 + 6006
= 51051.
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