In a class of 60 students, 29 students opted for Mathematics, 32 students opted for Biology, and 8 students opted for neither of these two subjects. How many students opted for both Mathematics and Biology? Choose the correct number of students.

Introduction / Context: This is a standard two-set counting problem using the inclusion–exclusion principle. It tests how to find the overlap (students who chose both subjects) when totals for each subject and “neither” are given.

Given Data / Assumptions:

• Total students = 60 • Mathematics (M) = 29 • Biology (B) = 32 • Neither M nor B = 8 • Required: M ∩ B

Concept / Approach: First find how many students took at least one of the two subjects: (M ∪ B) = Total − Neither. Then apply inclusion–exclusion: (M ∪ B) = M + B − (M ∩ B) So: (M ∩ B) = M + B − (M ∪ B).

Step-by-Step Solution: 1) Students who took at least one subject: M ∪ B = 60 − 8 = 52 2) Apply inclusion–exclusion: 52 = 29 + 32 − (M ∩ B) 3) Add: 29 + 32 = 61 4) Solve for intersection: M ∩ B = 61 − 52 = 9

Verification / Alternative check: If 9 students are in both, then only-Math = 29 − 9 = 20 and only-Biology = 32 − 9 = 23. Total with at least one = 20 + 23 + 9 = 52, which matches 60 − 8. Consistent.

Why Other Options Are Wrong: • 8 or 7: would make M ∪ B = 29 + 32 − overlap too large (53 or 54), contradicting 52. • 6 or 5: would make M ∪ B too large (55 or 56), again contradicting 52.

Common Pitfalls: • Adding 29 + 32 and forgetting to subtract overlap. • Using 8 as overlap instead of “neither”.

Final Answer: 9