Question

There are $3$ sections in a question paper and each section contains $5$ questions. A candidate has to answer a total of $5$ questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is:

• A. $2250$
• B. $2255$
• C. $1500$
• D. $3000$

Answer 1

The correct option is A

$2250$

Explanation for the correct answer

Different ways in which we can select the questions are in the following image.

One way is select two questions from two sets and one question from the third set.

We can select this type in $3$ ways.

The second way is select three questions from one set and one-one question from the remaining two sets.

We can select this type in $3$ ways.

Required number of ways,

$$\begin{gathered} =3\left({}^5{\mathrm{C}}_1\times {}^5{\mathrm{C}}_2\times {}^5{\mathrm{C}}_2\right)+3\left({}^5{\mathrm{C}}_1\times {}^5{\mathrm{C}}_1\times {}^5{\mathrm{C}}_3\right) \\ =3\left(5\times \frac{5!}{2!\left(5-2\right)!}\times \frac{5!}{2!\left(5-2\right)!}\right)+3\left(5\times 5\times \frac{5!}{3!\left(5-3\right)!}\right)\left[{}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}=\frac{\mathrm{n}!}{\mathrm{r}!\left(\mathrm{n}-\mathrm{r}\right)!}\right] \\ =3\left(5\times \frac{5!}{2!\times 3!}\times \frac{5!}{2!\times 3!}\right)+3\left(25\times \frac{5!}{3!\times 2!}\right) \\ =3\left(5\times 10\times 10\right)+3\left(25\times 10\right) \\ =2250 \end{gathered}$$

Hence, option A is correct .