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CAT 2022 Slot 1 — QA Question 20

Venn DiagramEasy

In a class of 100 students, 73 like coffee, 80 like tea and 52 like lemonade. It may be possible that some students do not like any of these three drinks. Then the difference between the maximum and minimum possible number of students who like all the three drinks is

Answer & solution

  • A

    52

  • 47

  • C

    48

  • D

    53

Solution

Easy

Use the "exactly one / exactly two / exactly three" counting identity a+2b+3c=sum of individual likesa+2b+3c=\text{sum of individual likes}. The maximum triple-overlap is capped by the smallest set; the minimum is forced because the total 205205 is too large to fit with c=0c=0.

1

Maximum possible "all three". No overlap can exceed the smallest individual group, which is lemonade =52=52:

cmax⁡=min⁡(73,80,52)=52\begin{aligned} &c_{\max}=\min(73,80,52)=52 \end{aligned}
2

Set up the overlap identity. With a,b,ca,b,c the counts liking exactly one, two, three drinks:

a+2b+3c=73+80+52=205\begin{aligned} &a+2b+3c=73+80+52=205 \end{aligned}
3

Push cc as low as possible. With 100100 students, even the most generous split with c=0c=0 falls short:

a=c=0, b=100 ⇒ a+2b+3c=200<205shifting one student from b to c adds +1 to the sum⇒ need to gain 5 ⇒ cmin⁡=5\begin{aligned} &a=c=0,\ b=100\ \Rightarrow\ a+2b+3c=200\lt 205\\ &\text{shifting one student from }b\text{ to }c\text{ adds }+1\text{ to the sum}\\ &\Rightarrow\ \text{need to gain }5\ \Rightarrow\ c_{\min}=5 \end{aligned}
4

Take the difference:

cmax⁡−cmin⁡=52−5=47\begin{aligned} &c_{\max}-c_{\min}=52-5=47 \end{aligned}

Difference =47=\mathbf{47}. Option (b).

CAT 2022 Slot 1 QA Q20: In a class of 100 students, 73 like coffee, 80 like tea and 52 like lemonade. It may be possible that some stu — Solution | TheCATExam