Chapter 9, Problem 14RE

Solution

To calculate: The numbers of different ways in which different employee representative committees can be formed if the committee must consists of 2 women and 2 men. There are total 35 employees, 21 women and 14 men.

The resultant answer is $19,110$ ways.

Given information:

There are total 35 employees, 21 women and 14 men.

Formula used: The number of combinations of n objects taken r at a time denoted by $C_{n_r}$ and given by $C_{n_r}=\frac{n!}{r!\left(n-r\right)!}$ .

Calculation:

There are total 21 women employees out of which 2 have to be selected. Then the expression for it will become: $C_{{21}_2}$

Rewrite the expression using the formula $C_{n_r}=\frac{n!}{r!\left(n-r\right)!}$ ,

  $\begin{array}{c}{C}_{{21}_2}=\frac{21!}{2!\left(21-2\right)!}\\ =\frac{21!}{2!19!}\end{array}$

Now expand the factorial,

  $\frac{21\cdot 20\cdot 19!}{2!19!}=210$

There are total 14 men employees out of which 2 have to be selected. Then the expression for it will become: $C_{{14}_2}$

  $\begin{array}{c}{C}_{{14}_2}=\frac{14!}{2!\left(14-2\right)!}\\ =\frac{14!}{2!12!}\\ =\frac{14\cdot 13\cdot 12!}{2!12!}\\ =91\end{array}$

Now the total number of ways in which 2 women and 2 men can be selected will be:

  $210\cdot 91=19,110$

Therefore, there are $19,110$ ways in which representative committee can be formed consisting of 2 women and 2 men.