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Difficulty: EasySet Theory Concepts and Venn Diagrams

A survey of 120120 university students evaluated course enrollment in Computer Science and Mathematics. Of the students surveyed, 7070 are enrolled in Computer Science, 6565 are enrolled in Mathematics, and 2525 are enrolled in neither course. How many students are enrolled in both Computer Science and Mathematics?

  1. A
    15
  2. B
    25
  3. 40Answer
  4. D
    50
  5. E
    95

Answer

40 students are enrolled in both Computer Science and Mathematics.
The correct answer is 40. From the total population of 120 students, 25 take neither subject, meaning 120−25=95120 - 25 = 95 students take at least one subject. According to the Principle of Inclusion-Exclusion, ∣C∪M∣=∣C∣+∣M∣−∣C∩M∣|C \cup M| = |C| + |M| - |C \cap M|. Substituting the known values yields 95=70+65−∣C∩M∣95 = 70 + 65 - |C \cap M|, which simplifies to 95=135−∣C∩M∣95 = 135 - |C \cap M|, giving ∣C∩M∣=40|C \cap M| = 40.

Step-by-Step Solution

1
Calculate the total number of students enrolled in at least one of the two courses.
Total in at least one course = 120−25=95120 - 25 = 95.
Subtracting the students enrolled in neither course from the total surveyed gives the union of the two sets, ∣C∪M∣|C \cup M|.
2
Apply the Principle of Inclusion-Exclusion for two sets.
∣C∪M∣=∣C∣+∣M∣−∣C∩M∣|C \cup M| = |C| + |M| - |C \cap M|, so 95=70+65−∣C∩M∣95 = 70 + 65 - |C \cap M|.
Adding ∣C∣|C| and ∣M∣|M| double-counts the students taking both courses, so subtracting the union yields the intersection.
3
Solve for the intersection ∣C∩M∣|C \cap M|.
∣C∩M∣=135−95=40|C \cap M| = 135 - 95 = 40.
Direct arithmetic evaluation gives the required count of students enrolled in both subjects.

Key Concept

Principle of Inclusion-Exclusion for Two Sets
Estimated Time:1m 0s
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Topics

Set Theory Concepts and Venn Diagrams

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