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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 x, then $x - 7$ is ........

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answer is 7.

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Detailed Solution

$S i n c e, n, (M \cup P \cup C) = 50 n (M) = 37 n (P) = 24, n (C) = 43 n (M \cap P) \leq 19 n (M \cap C) \leq 29 a n d n (P \cap C) \leq 20$

Since, we know that $n (P \cup C \cup M) = n (P) + n (C) + n (M)$

$- n (P \cap C) - n (C \cap M) - n (M \cap P) + n (P \cap C \cap M)$

$\Rightarrow 50 \geq 37 + 24 + 43 - n (M \cap P) - n (P \cap C) - n (M \cap C) + n (M \cap P \cap C)$

$\Rightarrow n (M \cap P \cap C) \leq n (M \cap P) + n (M \cap C) + n (P \cap C) - 54$

Therefore, the number of students is at most

$19 + 29 + 20 - 54 = 14$

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