Question

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

• A. \(30\)
• B. \(55\)
• C. \(35\)
• D. \(40\)

Hint:

For three-set Venn diagram problems:
Convert verbal relations about overlaps into equations using \(|A \cap B|\), \(|A \cap B \cap C|\), etc.
Use inclusion--exclusion to connect total strength with individual set sizes and intersections.
If asked for a maximum or minimum, express the required number in terms of a single variable and then optimize it under the given constraints.

Answer 1

The Correct Option is C

Approach: Build the seven-region Venn diagram explicitly and just demand every region be $\ge 0$; the divisibility and the answer both fall out.

Step 1: Regions. Let only-P $=p$, only-M $=m$, only-C $=k$, the (exclusive) pair-regions be $PM$, $PC$, $MC$, and the triple be $z$. The full pairwise overlaps are $PC{+}z = x$, $MC{+}z = x$, $PM{+}z = 2x$.

Step 2: "Physics not Maths" in region terms. That is $p + PC = p + (x - z)$. From $n(P) = p + PM + PC + z = 75$ and $PM = 2x - z$, $PC = x - z$, we get $p = 75 - 3x + z$. So Physics-not-Maths $= (75 - 3x + z) + (x - z) = 75 - 2x$ — same target, minimise $x$.

Step 3: Sum to 150 (= union). Adding all seven regions and using the three subject totals reproduces $4x = 76 + z$, so $x = (76 + z)/4$.

Step 4: Smallest legal $x$. Integrality forces $z \in \{4, 8, 12, \dots\}$. Take $z = 4 \Rightarrow x = 20$. Check regions: $p = 75 - 60 + 4 = 19$, $PM = 36$, $PC = MC = 16$, only-C $k = 40 - 16 - 16 - 4 = 4$, only-M $= 111 - 36 - 16 - 4 = 55$ — all non-negative, so this is feasible.

Step 5: Result. \[ 75 - 2(20) = 35. \] Maximum Physics-but-not-Maths $= 35$ (option c).