Out of 80 students who appeared for the school exams in Mathematics (M), Physics (P) and Chemistry (C), 50 passed M, 30 passed P and 40 passed C. At most 20 students passed M and P, at most 20…

At most $20$ students can pass all three, since every such student also passed Mathematics and Physics, whose intersection is capped at $20$. The bound is attained with $20$ passing all three and $30,10,20$ passing only Mathematics, only Physics and only Chemistry, respectively. This accounts for all $80$ students and each subject total, while $30$ in all three would exceed each pairwise cap.