In an examination, a question paper consists of 12 questions divided into two parts i.e., Part **I** and Part **II**, containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions?

#### Solution

It is given that the question paper consists of 12 questions divided into two parts – Part **I** and Part **II**, containing 5 and 7 questions, respectively.

A student has to attempt 8 questions, selecting at least 3 from each part.

This can be done as follows.

(a) 3 questions from part** I** and 5 questions from part **II**

(b) 4 questions from part **I** and 4 questions from part **II**

(c) 5 questions from part** I** and 3 questions from part **II**

3 questions from part** I** and 5 questions from part **II **can be selected in `""^5C_3 xx ""^7C_5`ways.

4 questions from part **I** and 4 questions from part **II** can be selected in `""^5C_4 xx ""^7C_4` ways.

5 questions from part **I** and 3 questions from part **II** can be selected in `""^5C_5 xx ""^7C_3` ways.

Thus, required number of ways of selecting questions