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,
$A \, B\, C$
Section $\begin{cases}2 & 2 & 1 \\2 & 1 & 2 \\1 & 2 & 2\end{cases}$
$\begin{cases}3 & 1 & 1 \\1 & 3 & 1 \\1 & 1 & 3\end{cases}$
$\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 $
$\Rightarrow \frac{4 !}{2 ! 2 !} \times \frac{4 !}{2 ! 2 !} \times \frac{4 !}{1 ! 3 !} \times 3+\frac{4 !}{3 ! 1 !} \times \frac{4 !}{1 ! 3 !} \times \frac{4 !}{1 ! 3 !} \times 3$
$\Rightarrow \,432+192=624$