A question paper is divided into two parts A and B and each part contains $5$ questions. The number of ways in which a candidate can answer $6$ questions selecting at least two questions from each part is?

The correct option is C

$200$

Explanation for the correct option:

Step 1: Find the number of ways of selecting $2$ questions from A and $4$ from B:

Find the number of ways of selecting $2$ questions from $5$ questions and $4$ from $5$ questions.

$\begin{array}{rcl}{n}_1& =& {}^{5}C_2·{}^{5}C_4\\ & =& 10·5\\ & =& 50\end{array}$

Step 2: Find the number of ways of selecting $3$ questions from A and $3$ from B:

Find the number of ways of selecting $3$ questions from $5$ questions and $3$ from $5$ questions.

$\begin{array}{rcl}{n}_2& =& {}^{5}C_3·{}^{5}C_3\\ & =& 10·10\\ & =& 100\end{array}$

Step 3: Find the number of ways of selecting $4$ questions from A and $2$ from B:

Find the number of ways of selecting $4$ questions from $5$ questions and $2$ from $5$ questions.

$\begin{array}{rcl}{n}_3& =& {}^{5}C_4·{}^{5}C_2\\ & =& 5·10\\ & =& 50\end{array}$

Step 4: Find the total number of ways:

Add up all the ways to get the total number of ways.

$$\begin{gathered} n=50+100+50 \\ =200 \end{gathered}$$

Hence, option (C) is the correct answer.