A cricket team of 11 players is to be selected from 16 players including
Question:

A cricket team of 11 players is to be selected from 16 players including 5 bowlers and 2 wicketkeepers. In how many ways can a team be selected so as to consist of exactly 3 bowlers and 1 wicketkeeper?

Solution:

There is a cricket team of 11 players is to be selected from 16 players, which must include 3 bowlers and a wicketkeeper.

$\Rightarrow$ There will be a team of 7 batsmen, 1 wicketkeeper and 3 bowlers.

$\Rightarrow$ There are 5 bowlers from which 3 is to be selected in ${ }^{5} C_{3}$ ways

$\Rightarrow$ There are two wicketkeepers out of which 1 is to be selected in ${ }^{2} \mathrm{C}_{1}$

$\Rightarrow$ Hence, from 9 players left 7 is to be selected from that in ${ }^{11} \mathrm{C}_{7}$ ways.

$\Rightarrow$ By Multiplication principle, we get

$={ }^{5} \mathrm{C}_{3} \times{ }_{2} \mathrm{C}_{1} \times{ }^{9} \mathrm{C}_{7}$

Applying ${ }^{n} C_{r}=\frac{n !}{r !(n-r) !}$

$=720$ ways