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Difficulty: MediumSets and Set Operations

In a secondary school class of 45 students, 28 study Chemistry, 25 study Physics, and 6 study neither of the two subjects. How many students study both Chemistry and Physics?

Answer: 14 / 14 students

Answer

14 students study both Chemistry and Physics.
First, find the number of students taking at least one subject by subtracting those taking neither from the total: 45 - 6 = 39. Then, apply the formula n(C ∪ P) = n(C) + n(P) - n(C ∩ P). Substituting the values gives 39 = 28 + 25 - n(C ∩ P), which simplifies to n(C ∩ P) = 53 - 39 = 14.

Step-by-Step Solution

1
Calculate the number of students taking at least one of the two subjects.
n(Chemistry ∪ Physics) = 45 - 6 = 39
Students who study neither subject must be subtracted from the total class population to find the cardinality of the union.
2
Set up the two-set inclusion-exclusion formula.
n(Chemistry ∪ Physics) = n(Chemistry) + n(Physics) - n(Chemistry ∩ Physics)
The sum of individual set cardinalities overcounts elements present in both sets.
3
Substitute the known values into the equation and solve for the intersection.
39 = 28 + 25 - n(Chemistry ∩ Physics) ⇒ n(Chemistry ∩ Physics) = 53 - 39 = 14
Subtracting 39 from 53 gives the exact number of students taking both subjects.

Key Concept

Two-set inclusion-exclusion principle and complement of a set
Estimated Time:1m 30s
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Topics

Sets and Set Operations

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