CAT 2025

Slot 3

QA

Question & SolutionQues & Sol

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. Concept Used

• Topic: Venn Diagrams — Sets, Inclusion-Exclusion Principle, and Maximisation

• Concept:

The sum of all the entries in the Venn diagram must equal the total number of students, which is 150.

2. Calculation

We know that the sum of all entries in the Venn diagram should be 150. Therefore,

$$ 111+75-2k+40-(k-x)-k+0=150 $$

Simplifying,

$$ 76=4k-x $$

where $x\geq1$.

We need to maximise the number of students who chose Physics but not Mathematics. This number is:

$$ 75-2k $$

To maximise this value, we must minimise $k$.

From the equation,

$$ 4k=76+x $$

Therefore,

$$ k=19+\frac{x}{4} $$

Since $x$ and $k$ are both integers, $x$ must be divisible by 4.

To minimise $k$, we take the minimum possible value of $x$:

$$ x=4 $$

Therefore,

$$ k=19+\frac{4}{4}=20 $$

Thus, the maximum possible number of students who chose Physics but not Mathematics is:

$$ 75-2\times20=35 $$

3. Solution

Answer = 35 ✅

The maximum possible number of students who chose Physics but not Mathematics is 35.