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 passed in all the three subjects is ________.

Answer 1

Solution: Using the principle of inclusion-exclusion for three sets $M$, $P$, and $C$, we have:

$|M \cup P \cup C| = |M| + |P| + |C| - |M \cap P| - |P \cap C| - |M \cap C| + |M \cap P \cap C|$

Given:

• $|M| = 20$
• $|P| = 25$
• $|C| = 16$
• $|M \cap P| \leq 11$
• $|P \cap C| \leq 15$
• $|M \cap C| \leq 10$

Since $|M \cup P \cup C| = 40$, substitute the values and solve for $|M \cap P \cap C|$:

$40 = 20 + 25 + 16 - 11 - 15 - 10 + x$

$x = 10$

Thus, the maximum number of students who passed in all three subjects is 10.