# 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?

A student has to attempt 8 questions in total and at least 3 from each part.

There can be three cases:

Case 1: 3 from part 1 and 5 from part 2

5C3 × 7C5

= 10× 21

= 210 ways

Case 2: 4 from part 1 and 4 from part 2

5C4 × 7C4

= 5× 35

= 175 ways

Case 3: 5 from part 1 and 3 from part 2.

5C5 × 7C3

= 35 ways

So total ways = 210 + 175 + 35

= 420 ways

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