Question

There are three sections in a question paper, each containing $5$ questions. A candidate has to solve any $5$ questions, choosing at least one from each section. Find the number of ways in which the candidate can choose the questions.

• A. ${}^{15}\mathrm{C}_5$
• B. $2250$
• C. ${}^3\mathrm{C}_1\times {}^{12}\mathrm{C}_4$
• D. None of these.

Answer 1

The correct option is B

$2250$

Explanation for correct options

Given: There are three sections containing $4,5$ and $6$ questions respectively.

Now, from each question, $3$ questions should be answered.

Therefore total number of ways, for the selection of question can be done

$$\begin{gathered} (1,1,3)+(1,2,2)+(1,3,1)+(2,1,2)+(2,2,1)+(3,1,1) \\ ={}^5\mathrm{C}_1\times {}^5\mathrm{C}_1\times {}^5\mathrm{C}_3+{}^5\mathrm{C}_1\times {}^5\mathrm{C}_2\times {}^5\mathrm{C}_2+{}^5\mathrm{C}_1\times {}^5\mathrm{C}_3\times {}^5\mathrm{C}_1+{}^5\mathrm{C}_2\times {}^5\mathrm{C}_1\times {}^5\mathrm{C}_2+{}^5\mathrm{C}_2\times {}^5\mathrm{C}_2\times {}^5\mathrm{C}_1+{}^5\mathrm{C}_3\times {}^5\mathrm{C}_1\times {}^5\mathrm{C}_1 \\ =5\times 5\times 10+5\times 10\times 10+5\times 10\times 5+10\times 5\times 10+10\times 10\times 5+10\times 5\times 5 \\ =2250 \end{gathered}$$

Hence option $\left(\mathrm{B}\right)$, $2250$ is the correct answer.