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 ?

  • 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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