Among 22 cricket players, there are 3 wicketkeepers and 6 bowlers. In how many ways can a team of 11 players be chosen so as to include exactly one wicketkeeper and atleast 4 bowlers?
Answers
Answer:
1,02,531
Step-by-step explanation:
Hi,
Given that among 22 cricket players , there are 3 wicket
keepers and 6 bowlers . Required to find the number of ways
can a team of 11 players be chosen so as to include exactly one
wicket keeper and at least 4 bowlers
We can divide the above problem into following cases:
Case 1: 1 wicket keeper + 4 bowlers + 6 other players
1 wicket keeper among 3 can be selected in 3 ways
4 bowlers out of 6 can be selected in ⁶C₄ ways = 15 ways
Rest of 6 players out of 13 can be selected in ¹³C₆ ways = 1716
ways,
total ways this combination of selection can be made are
3*15*1716 = 77220 ways
Case 2: 1 wicket keeper + 5 bowlers + 5 other players
1 wicket keeper among 3 can be selected in 3 ways
5 bowlers out of 6 can be selected in ⁶C₅ ways = 6 ways
Rest of 5 players out of 13 can be selected in ¹³C₅ ways = 1287
ways
total ways this combination of selection can be made are
3*6*1287 = 23166 ways
Case 3: 1 wicket keeper + 6 bowlers + 4 other players
1 wicket keeper among 3 can be selected in 3 ways
6 bowlers out of 6 can be selected in only 1 way
Rest of 4 players out of 13 can be selected in ¹³C₄ ways = 715
ways
total ways this combination of selection can be made are
3*1*715 = 2145 ways .
Hence, total number of ways a team of 11 can be formed are
(77220 + 23166 + 2145) = 1,02,531 ways.
Hope, this helps !
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