Question

There are three sections in a question paper, each containing $4$ questions. If a candidate has to answer only $5$ questions from this paper without leaving any section, then number of ways the candidate can make the choice of questions is

• A. 624
• B. 704
• C. 384
• D. 432

Answer 1

Answer: (A)

Number of ways to select 5 questions from these sections as follows,
$ABC$
Section $\{\begin{array}{llc}2& 2& 1\\ 2& 1& 2\\ 1& 2& 2\end{array}$
$\{\begin{array}{llc}3& 1& 1\\ 1& 3& 1\\ 1& 1& 3\end{array}$
$\Rightarrow {}^4{C}_2\times {}^4{C}_2\times {}^4{C}_1\times 3+{}^4{C}_3\times {}^4{C}_1\times {}^4{C}_1\times 3$
$\Rightarrow \frac{4!}{2!2!}\times \frac{4!}{2!2!}\times \frac{4!}{1!3!}\times 3+\frac{4!}{3!1!}\times \frac{4!}{1!3!}\times \frac{4!}{1!3!}\times 3$
$\Rightarrow 432+192=624$