Two-language enrollment using inclusion–exclusion: In a school of 600 students, each offered English or Hindi or both. If 75% took English and 45% took Hindi, how many offered both subjects?

Introduction / Context:
Classic set problems about course enrollment are solved with inclusion–exclusion. Given totals for English and Hindi and the universe size, we can compute the overlap directly.

Given Data / Assumptions:

• Total students U = 600
• n(English) = 75% of 600 = 450
• n(Hindi) = 45% of 600 = 270
• All students chose at least one of the two languages (as stated)

Concept / Approach:
Inclusion–exclusion for two sets: n(E ∪ H) = n(E) + n(H) − n(E ∩ H). Here n(E ∪ H) = U, so solve for n(E ∩ H).

Step-by-Step Solution:
n(E ∩ H) = n(E) + n(H) − U= 450 + 270 − 600= 120

Verification / Alternative check:
Unique-English = 450 − 120 = 330; Unique-Hindi = 270 − 120 = 150; 330 + 150 + 120 = 600 (balanced).

Why Other Options Are Wrong:
48, 60, 80 do not satisfy the totals when back-checked; 140 would overcount overlap and push the union below 600.

Common Pitfalls:
Adding 75% and 45% to 120% and forgetting to subtract the overlap once; assuming disjoint groups when the problem states “or both.”

Final Answer:
120