CAT 2025 Slot 3 · Quant · Question 14

CAT 2025 Slot 3 previous year question on Set Theory and Venn Diagrams, with the answer and a step-by-step solution.

Set Theory and Venn DiagramsModern MathsMultiple choiceHard
In a class of 150 students, 75 students chose physics, 111 students chose mathematics and 40 students chose chemistry. All students chose at least one of the three subjects and at least one student chose all three subjects. The number of students who chose both physics and chemistry is equal to the number of students who chose both chemistry and mathematics, and this is half the number of students who chose both physics and mathematics. The maximum possible number of students who chose physics but not mathematics, is
  1. A
    30
  2. B
    35
  3. C
    40
  4. D
    55
Pick an option, then check.
Answer & solution

Answer: Option B

Write x for the number choosing physics and chemistry, which also equals the number choosing chemistry and mathematics, and 2x for the number choosing physics and mathematics. Let t be the number choosing all three.

Every student chose at least one subject, so the three sets cover all 150.

By inclusion and exclusion, 75+111+40-2x-x-x+t=150.

That is 226-4x+t=150, so t=4x-76.

The number choosing physics but not mathematics is 75-2x, which grows as x shrinks — so minimise x.

At least one student chose all three, so t\ge 1, giving 4x-76\ge 1 and x\ge 19.25.

Since x counts students it is a whole number, so the smallest allowed value is x=20, giving t=4.

Then the answer is 75-2\left( 20 \right)=35.

Checking the seven regions at x=20, t=4: physics only 19, maths only 55, chemistry only 4, physics-maths only 36, physics-chemistry only 16, maths-chemistry only 16 — all non-negative and totalling 150.

Shortcut. Inclusion and exclusion collapses the whole problem to the single relation t=4x-76. Since the target 75-2x falls as x rises, the answer sits at the smallest x that keeps t a positive whole number — one inequality, no case work.

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