Question

Out of a group of 50 students taking examinations in Mathematics, Physics, and Chemistry, 37 students passed Mathematics, 24 passed Physics, and 43 passed Chemistry. Additionally, no more than 19 students passed both Mathematics and Physics, no more than 29 passed both Mathematics and Chemistry, and no more than 20 passed both Physics and Chemistry. What is the maximum number of students who could have passed all three examinations?

• A. 12
• B. 9
• C. 14
• D. 10

Hint:

Use the inclusion-exclusion principle to maximize or minimize the number of students in multiple sets.

Answer 1

The Correct Option is C

We need to find the maximum number of students who could have passed all three subjects. Let:
\( M \) be the set of students who passed Mathematics,
\( P \) be the set of students who passed Physics,
\( C \) be the set of students who passed Chemistry.
We are given:
\( |M| = 37 \),
\( |P| = 24 \),
\( |C| = 43 \),
\( |M \cap P| \leq 19 \),
\( |M \cap C| \leq 29 \),
\( |P \cap C| \leq 20 \).
To find the maximum number of students who passed all three subjects, we use the inclusion-exclusion principle. The maximum number of students who passed all three subjects occurs when the pairwise intersections are as large as possible, i.e., when: \[ |M \cap P \cap C| = \boxed{14}. \] Thus, the correct answer is: \[ \boxed{14}. \]