Given 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers

There are 5 players how bowl, and we can require 4 bowlers in a team of 11

Number of ways in which bowlers can be selected are: ^{5}C_{4}

Now other players left are = 17 – 5(bowlers) = 12

Since we need 11 players in a team and already 4 bowlers are selected, we need to select 7 more players from 12.

Number of ways we can select these players are: ^{12}C_{7}

Total number of combinations possible are: ^{5}C_{4} × ^{12}C_{7}

^{5}C_{4} × ^{12}C_{7 }= \( \dfrac{5!}{4!(5-4)!}\times \dfrac{12!}{7!(12-7)!}\)

= \( \dfrac{5!}{4!\times 1!}\times \dfrac{12!}{7!\times 5!}\)

^{5}C_{4} × ^{12}C_{7 =} \( \dfrac{5\times 4!}{1!\times 4!}\times \dfrac{12\times 11\times 10\times 9\times 8\times 7!}{5!\times 7!}\)

= \( \dfrac{5}{1}\times \dfrac{95040}{120}\)

= \( 5\times 792=3960\)

Number of ways we can select a team of 11 players where 4 players are bowlers from 17 players are 3960

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