Class 11

Math

Algebra

Permutations and Combinations

In how many ways can one select a cricket team of eleven from $17$ players in which only $5$ players can bowl if each cricket team of $11$ must include exactly $4$ bowlers?

A cricket team of $$11$$ players is to be selected in such a way that there are exactly $$4$$ bowlers.

$$4$$ bowlers can be selected in $$^5C_4$$ways and the remaining $$7$$ players can be selected out of the $$12$$ players in $$^{12}C_7$$ ways.

Thus, by multiplication principle, required number of ways of selecting cricket team

$$\Rightarrow \displaystyle ^{5}C_4 \times ^{12}C_7=\frac {5!}{4!1!}\times \frac {12!}{7!5!}$$

$$=\displaystyle \frac {12\times 11\times 10\times 9\times 8}{5\times 4\times 3\times 2\times 1}\times 1$$

$$=3960$$