Question

Out of $5$ nurses and $2$ doctors, a commitee of $3$ members is to be formed. In how many ways can it be done if at least one doctor is to be included?

• A. $5$
• B. $15$
• C. $25$
• D. $50$

Answer 1

The correct option is C $25$

Number of nurses $=5$, Number of doctors $=2$. A committee of $3$ is to be formed.

The number of ways to form a committee of $3$ members in such a way to have at least one doctor $=2$ nurses out of $5$ and $1$ doctor out of 2 $\left({{}^{5}C}_{2}and{{}^{2}C}_1\right)$ or 1 nurse out of $5$ nurses and $2$ doctors out of $2$ doctors $\left({{}^{5}C}_{1}and{{}^{2}C}_2\right)$

Thus, the total number of ways to form the committee

$=[{{}^{5}C}_2\times {{}^{2}C}_1]+[{{}^{5}C}_1\times {{}^{2}C}_2]$

$\Rightarrow \left[\frac{5!}{2!3!}\times \frac{2!}{1!1!}\right]+\left[\frac{51}{4!1!}\times \frac{2!}{2!0!}\right]=(10\times 2)+(5\times 1)=25$