Question

A survey of 450 students about their subjects of interest resulted in the following outcome.


150 students are interested in Mathematics.
200 students are interested in Physics.
175 students are interested in Chemistry.
50 students are interested in Mathematics and Physics.
60 students are interested in Physics and Chemistry.
40 students are interested in Mathematics and Chemistry.
30 students are interested in Mathematics, Physics and Chemistry.
Remaining students are interested in Humanities. Based on the above information, the number of students interested in Humanities is:

• A. 10
• B. 30
• C. 40
• D. 45

Hint:

Use the principle of inclusion and exclusion to calculate the total number of students interested in at least one subject, and then subtract from the total number of students.

Answer 1

The Correct Option is D

We can use the principle of inclusion and exclusion to solve this problem. Let:
- \( M \) be the set of students interested in Mathematics,
- \( P \) be the set of students interested in Physics,
- \( C \) be the set of students interested in Chemistry.
The total number of students interested in at least one of the three subjects is given by: \[ |M \cup P \cup C| = |M| + |P| + |C| - |M \cap P| - |P \cap C| - |M \cap C| + |M \cap P \cap C| \] Substitute the given values: \[ |M \cup P \cup C| = 150 + 200 + 175 - 50 - 60 - 40 + 30 \] Simplifying: \[ |M \cup P \cup C| = 405 \] Thus, the number of students who are interested in at least one subject is 405. The total number of students is 450, so the number of students interested in Humanities is: \[ 450 - 405 = 45 \] Thus, the number of students interested in Humanities is 45.