From 50 students taking examinations in Mathematics, Physics and Chemistry, 37 passed Mathematics, 24 Physics and 43 Chemistry. At most 19 passed Mathematics and Physics, at most 29 Mathematics and Chemistry and at most 20 Physics and Chemistry. The largest possible number that could have passed all three exams is

A
10
B
12
C
9
D
none of these

Solution:

$\begin{array}{l} n (M \cup P \cup C) = 50, n (M) = 37, n (P) = 24, n (C) = 43, n (M \cap P) \leq 19, n (M \cap C) \leq 29, n (C \cap P) \leq 20 \\ n (M \cup P \cup C) = n (M) + n (P) + n (C) - n (M \cap P) - n (C \cap P) - n (M \cap C) + n (M \cap P \cap C) \\ \Rightarrow n (M \cap P \cap C) \leq n (M \cap P) + n (M \cap C) + n (P \cap C) - 54 \end{array}$

Therefore, the number of students that could have passed all three exams is at most 19 + 29 + 20 - 54 = 14

Solution:

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