Question:
500 students are taking one or more courses out of Chemistry, Physics and Mathematics. Registration records indicate course enrolment as follows : Chemistry (329), Physics (186), Mathematics (295), Chemistry and Physics (83), Chemistry and Mathematics (217) and Physics and Mathematics (63). How many students are taking all 3 subjects ?
500 students are taking one or more courses out of Chemistry, Physics and Mathematics. Registration records indicate course enrolment as follows : Chemistry (329), Physics (186), Mathematics (295), Chemistry and Physics (83), Chemistry and Mathematics (217) and Physics and Mathematics (63). How many students are taking all 3 subjects ?
Updated On: Aug 18, 2025
- 37
- 53
- 47
- 43
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The Correct Option is B
Solution and Explanation
To determine how many students are taking all three subjects—Chemistry, Physics, and Mathematics—we will use the principle of inclusion-exclusion. Let's define the following sets:
- C: Students taking Chemistry
- P: Students taking Physics
- M: Students taking Mathematics
Given data:
- |C| = 329
- |P| = 186
- |M| = 295
- |C ∩ P| = 83
- |C ∩ M| = 217
- |P ∩ M| = 63
We need to find |C ∩ P ∩ M|, the number of students taking all three subjects. The formula for inclusion-exclusion for three sets is:
|C ∪ P ∪ M| = |C| + |P| + |M| - |C ∩ P| - |C ∩ M| - |P ∩ M| + |C ∩ P ∩ M|
It is given that the total number of students is 500, thus:
500 = 329 + 186 + 295 - 83 - 217 - 63 + |C ∩ P ∩ M|
Calculate the sum on the right:
500 = 810 - 363 + |C ∩ P ∩ M|
500 = 447 + |C ∩ P ∩ M|
Solving for |C ∩ P ∩ M| gives:
|C ∩ P ∩ M| = 500 - 447
|C ∩ P ∩ M| = 53
Therefore, 53 students are taking all three subjects.
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