Question

An Engineer Association consists of $5$ civil engineers and $5$ mechanical engineers.

In how many ways a committee of $3$ civil engineers and $2$ mechanical engineers be appointed $?$

Answer 1

Find the number of ways to appoint a committee of $3$ civil engineers and $2$ mechanical engineers:

Step-1: $3$ civil engineers are to be appointed from $5$ civil engineers:

$\begin{array}{rcl}{}^5{\mathrm{C}}_3& =& \left(5\times \frac{4}{2}\right)\\ & =& \left(\frac{20}{2}\right)\\ & =& 10\mathrm{ways}\end{array}$

Step-2: $2$ mechanical engineers are to be appointed from $5$ mechanical engineers:

$$\begin{gathered} {}^5{C}_2=\left(5\times \frac{4}{2}\right) \\ =\frac{20}{2} \\ =10\mathrm{ways} \end{gathered}$$

Step-3: Number of ways a committee of $3$ civil engineers and $2$ mechanical engineers be appointed:

$$\begin{gathered} =10\times 10 \\ =100\mathrm{ways} \end{gathered}$$ (Find the product of above steps to get the combined probability)

Hence, a committee of $3$ civil engineers and $2$ mechanical engineers can be appointed in $\mathbf{100}\mathbf{}\mathbf{ways}$ .