In a group of 100 students, each student has to opt for one or more of the three subjects among Physics, Chemistry and Mathematics. The number of students who opted for Mathematics is more than the number of students who opted for Physics, which, in turn, is more than the number of students who opted for Chemistry, which, in turn, is more than the number of students who opted for exactly two of the three subjects, which, in turn, is more than the number of students who opted for all the three subjects. It is also known that at least one student opted for all the three subjects.
- What can be the maximum number of students who opted for Chemistry?
- What can be the minimum number of students who opted for Mathematics?
- What can be the maximum number of students who opted for only Physics?
- What can be the maximum number of students who opted for Physics and Chemistry but not Mathematics?
Best Answer
Let the set of mathematics students be $M$, and $P$ for the physics students, $C$ for chemistry. By a standard counting theorem:
$$ 100 = |C \cup M \cup M| = |M| + |P| + |C| - |M \cap P| - |M \cap C| - |P \cap C| + |M \cap P \cap C|$$
Now $|M| > |P| > |C| > (|M \cap P| + |M \cap C| + |P \cap C| - 3|M \cap P \cap C|)$, where the last counts the number of students that have exactly chosen two out of three. We also know that $|M \cap P \cap C| \ge 1$. Drawing the Venn diagram might help.