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?

Maximum Students Passing All Three Exams

Given:

• Total students = 50
• \( |M| = 37 \), \( |P| = 24 \), \( |C| = 43 \)
• \( |M \cap P| \leq 19 \), \( |M \cap C| \leq 29 \), \( |P \cap C| \leq 20 \)

We use the inclusion-exclusion principle:

\[ |M \cup P \cup C| = |M| + |P| + |C| - |M \cap P| - |M \cap C| - |P \cap C| + |M \cap P \cap C| \]

Let \( x = |M \cap P \cap C| \). Then:

\[ 50 \geq 37 + 24 + 43 - 19 - 29 - 20 + x \Rightarrow 50 \geq 36 + x \Rightarrow x \leq 14 \]

✅ Final Answer: \(\boxed{14}\)