A question paper is divided into two parts A and B and each part contains 5 questions. In how many ways a student can answer the question paper, if he has to solve total 6 questions including atleast 2 from each section.

Question

I solved the question using the logic first select two questions from each sections. ($\binom{5}{2} * \binom{5}{2}$). Then from remaining 6 questions choose any 2. therefore final ans is = $\binom{5}{2} * \binom{5}{2} * \binom{6}{2}$ which is 1500. Is my answer correct. If wrong What is the mistake. I found this question in an aptitude book which I have. The book mentions the answer as 200.

Answer 1

In your method you’re doing repeated counting. i.e., in there are common selections in the initial ${}^5C_2$ and the last ${}^6C_2.$

You can do this problem as follows.

Since we need to select $6$ questions with minimum $2$ each from the two parts, total possibilities are

1. $3$ each from the two parts: $2 \times {}^6C_3= 100.$
2. $4$ from part $A$ and $2$ from part $B = {}^5C_4 \times {}^5C_2 = 50.$
3. $2$ from part $A$ and $4$ from part $B = {}^5C_2 \times {}^5C_4 = 50.$

These possibilities are mutually exclusive (no overlapping) and exhaustive (all cases covered). So, total possibilies $ = 100+50+50 = 200.$