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CAT 2020 Slot 2 — QA Question 20

Venn DiagramEasy

Students in a college have to choose at least two subjects from chemistry, mathematics and physics. The number of students choosing all three subjects is 18, choosing mathematics as one of their subjects is 23 and choosing physics as one of their subjects is 25. The smallest possible number of students who could choose chemistry as one of their subjects is

Answer & solution

  • A

    21

  • 20

  • C

    19

  • D

    22

Solution

Easy

Everyone takes at least two subjects, so the only regions are the three "exactly two" pairs and the "all three" centre (=18=18). Use the Maths and Physics totals to push as many students as possible away from Chemistry, then whatever Physics students are forced to also take Chemistry give the minimum.

Regions (no single-subject students): let MPMP = only Maths+Physics, MCMC = only Maths+Chem, PCPC = only Phys+Chem, and centre (all three) =18=18. Maths total =MP+MC+18=23=MP+MC+18=23; Physics total =MP+PC+18=25=MP+PC+18=25.

1

Maximize the non-Chemistry overlap. To minimize Chemistry we want MPMP (Maths+Physics only, no Chemistry) as large as possible. From the Maths equation:

MP+MC=23−18=5⇒ MP≤5(take MC=0, MP=5)\begin{aligned} &MP + MC = 23 - 18 = 5\\ &\Rightarrow\ MP \le 5 \quad\text{(take }MC=0,\ MP=5\text{)} \end{aligned}
2

Force the remaining Physics students into Chemistry. From the Physics equation with MP=5MP=5:

PC=25−18−MP=25−18−5=2\begin{aligned} &PC = 25 - 18 - MP = 25 - 18 - 5 = 2 \end{aligned}

These 22 students take Physics together with Chemistry.

3

Count Chemistry. Chemistry students =MC+PC+centre= MC + PC + \text{centre}. With MC=0MC=0, PC=2PC=2, centre =18=18:

Chemistry=0+2+18=20\begin{aligned} &\text{Chemistry} = 0 + 2 + 18 = 20 \end{aligned}
Smallest possible Chemistry count=20\text{Smallest possible Chemistry count} = 20
CAT 2020 Slot 2 QA Q20: Students in a college have to choose at least two subjects from chemistry, mathematics and physics. The number — Solution | TheCATExam