**Question:**

In how many ways can a cricket team be selected from a group of 25 players containing 10 batsmen, 8 bowlers, 5 all-rounders and 2 wicketkeepers, assuming that the team of 11 players requires 5 batsmen, 3 all-rounders, 2 bowlers and 1 wicketkeeper?

**Solution:**

A team of 11 players is to be made from 25 players.

$\Rightarrow$ Selecting 5 batsmen from 10 in ${ }^{10} \mathrm{C}_{5}$ ways.

$\Rightarrow$ Selecting 3 all-rounders from 5 in ${ }^{5} \mathrm{C}_{3}$ ways.

$\Rightarrow$ Selecting 2 bowlers from 8 in ${ }^{8} \mathrm{C}_{2}$ ways.

$\Rightarrow$ Selecting 1 wicketkeeper from 2 in ${ }^{2} C_{1}$ ways.

Thus, by the multiplication principle, we get

$={ }^{10} C_{5} \times{ }^{8} C_{2} \times{ }^{5} C_{3} \times{ }^{2} C_{1}$ ways

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

= 141120 ways