Inclusion–exclusion with pass percentages: 49% failed in English, 36% failed in Hindi, and 15% failed in both. If 450 students passed overall, how many students appeared for the exam in total?

Introduction / Context:
When two failure sets overlap, inclusion–exclusion helps compute the union (failed in at least one subject). Passing percentage is then 100% minus this union. Knowing the count of passed students lets us back-solve for the total cohort.

Given Data / Assumptions:

• Failed English = 49%.
• Failed Hindi = 36%.
• Failed both = 15%.
• Passed students count = 450.

Concept / Approach:
Inclusion–exclusion for failure: Fail(any) = 49% + 36% − 15% = 70%. Therefore, Pass = 30% of total. If 30% equals 450, total = 450 / 0.30.

Step-by-Step Solution:

Verification / Alternative check:
30% of 1500 is 450; remaining 1050 students failed at least one, consistent with the 70% union of failures.

Why Other Options Are Wrong:

• 2000, 1800, 1100: 30% of these do not equal 450, hence they do not satisfy the given pass count.

Common Pitfalls:
Adding the failure rates without subtracting the overlap, which would incorrectly give 85% failure and 15% pass.

Final Answer: