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CAT 2019 Slot 1 — QA Question 30

Venn DiagramEasy

A club has 256 members of whom 144 can play football, 123 can play tennis, and 132 can play cricket. Moreover, 58 members can play both football and tennis, 25 can play both cricket and tennis, while 63 can play both football and cricket. If every member can play at least one game, then the number of members who can play only tennis is

Answer & solution

  • A

    45

  • 43

  • C

    32

  • D

    38

Solution

Easy

Everyone plays at least one game, so the union equals 256. Use inclusion–exclusion to find the triple-overlap n(F∩T∩C)n(F\cap T\cap C), then subtract the two-game overlaps (each minus the triple) from n(T)n(T) to leave players of only tennis.

n(F∪T∪C)=256n(F{\cup}T{\cup}C)=256, n(F)=144, n(T)=123, n(C)=132n(F)=144,\ n(T)=123,\ n(C)=132, n(F∩T)=58, n(C∩T)=25, n(F∩C)=63n(F\cap T)=58,\ n(C\cap T)=25,\ n(F\cap C)=63.

1

Inclusion–exclusion for the triple overlap.

n(F∪T∪C)=∑n−∑pairs+n(F∩T∩C)⇒ 256=(144+123+132)−(58+25+63)+t(t=n(F∩T∩C))⇒ 256=399−146+t ⇒ t=3\begin{aligned} &n(F{\cup}T{\cup}C)=\textstyle\sum n-\sum\text{pairs}+n(F\cap T\cap C)\\ &\Rightarrow\ 256=(144+123+132)-(58+25+63)+t\quad\text{(}t=n(F\cap T\cap C))\\ &\Rightarrow\ 256=399-146+t\ \Rightarrow\ t=3 \end{aligned}
2

Only-tennis count. From n(T)n(T) remove the two tennis pair-regions and add back the triple once (it was removed twice).

only T=n(T)−(n(F∩T)+n(C∩T))+n(F∩T∩C)⇒ =123−(58+25)+3(from step 1, t=3)⇒ =123−83+3=43\begin{aligned} &\text{only }T=n(T)-\big(n(F\cap T)+n(C\cap T)\big)+n(F\cap T\cap C)\\ &\Rightarrow\ =123-(58+25)+3\quad\text{(from step 1, }t=3)\\ &\Rightarrow\ =123-83+3=43 \end{aligned}
Only tennis=43\text{Only tennis}=43
CAT 2019 Slot 1 QA Q30: A club has 256 members of whom 144 can play football, 123 can play tennis, and 132 can play cricket. Moreover, — Solution | TheCATExam