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, 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 ⁵C₃ × ⁷C₅ ways.
4 questions from Part I and 4 questions from Part II can be selected in ⁵C₄ × ⁷C₄ ways.
5 questions from Part I and 3 questions from Part II can be selected in ⁵C₅ × ⁷C₃ ways.

Thus, required number of ways of selecting questions = 
⁵C₃ × ⁷C₅ + ⁵C₄ × ⁷C₄ + ⁵C₅ × ⁷C₃

= (10 × 21) + (5 × 35) + (1 × 35) = 210 + 175 + 35 = 420

Hence, the required number of ways = 420.