Question

A law firm has eleven male and ten female lawyers. A committee of five, two male lawyers and three female lawyers, is to be chosen to represent the firm at a conference. How many committees are possible?

Answer 1

Making sellection of a smaller group from a larger group is doe by combinations.
Here, two male lawyers are to be chosen from eleven male lawyers and three female lawyers are to be chosen from ten female lawyers.

The numbers of ways to choose r objects from n objects is given by
$^nC_r= \frac{n!}{(n-r)!r!}$

Thus,
$^{11}C_2= \frac{11!}{(11-2)!\times2!} = \frac{11!}{9!\times2!} = \frac{11\times10\times9!}{9!\times2\times1} =11\times5=55$

Also,
$^{10}C_3= \frac{10!}{(10-3)!\times3!} = \frac{10!}{7!\times3!} = \frac{10\times9\times8\times7!}{7!\times3\times2\times1} =10\times4\times3=120$

Therefore, possible number of committes = 55 x 120 = 6,600 committes.