Question

A committee of $3$ persons is to be constituted from a group of $2$ men and $3$ women. In how many ways can this be done? How many of these committees would consist of $1$ man and $2$ women?

Answer 1

Selecting a commitee of $3$ persons from $5$ different persons.
Given $n=5$ and $r=3$
$\therefore$ Required number of ways $={}^n{C}_r={}^5{C}_3$
$=\frac{5!}{3!(5-3)!}=\frac{5\times 4\times 3!}{3!\times 2\times 1}=\frac{20}{2}=10$

Number of ways of selecting a commitee consisting of $1$ man and $2$ women

Now, selecting $1$ man from $2$ man
So $n=2,r=1$
$\therefore$ Number of ways $={}^n{C}_r{=}^2{C}_1=2$

Now, selecting $2$ women from $3$ women
So, $n=3$ and $r=2$
$\therefore$ Number of ways $={}^n{C}_r{=}^3{C}_2$
$=\frac{3!}{(3-2)!2!}=3$

$\therefore$ Number of ways of selecting a commitee of $1$ man and $2$ women $=2\times 3=6$ ways.