A group of 40 students appeared in an examination of 3 subjects --- Mathematics, Physics &…

Question

A group of 40 students appeared in an examination of 3 subjects --- Mathematics, Physics \& Chemistry. It was found that all students passed in at least one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, at most 11 students passed in both Mathematics and Physics, at most 15 students passed in both Physics and Chemistry, at most 15 students passed in both Mathematics and Chemistry. The maximum number of students who passed in all the three subjects is \_\_\_\_.

Answer: 10

Accepted Answer

Let $x = n(M\cap P\cap C)$ and $S = n(M\cap P)+n(P\cap C)+n(M\cap C)$. By inclusion--exclusion, since $n(M\cup P\cup C)=40$:
$$40 = 20+25+16 - S + x \implies S = 21+x$$

Since $x \leq n(M\cap P)$, $x\leq n(P\cap C)$, $x\leq n(M\cap C)$ individually, summing gives $S \geq 3x$. Combined with $S=21+x$:
$$21+x \geq 3x \implies 21\geq2x \implies x\leq10.5 \implies x\leq10$$

This bound is achievable (with $n(M\cap P)=11, n(P\cap C)=15, n(M\cap C)=15$ suitably arranged), so the maximum number of students who passed all three subjects is 10.

Solutions · 1