CAT 2019 Set-1 | Question: 79

Question

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 

• A. $45$
• B. $38$
• C. $32$
• D. $43$

Answer 1

We can draw the Venn diagram,

Given that,

• $ n (\text{F} \cup \text{T} \cup \text{C}) = 256 $
• $ n (\text{F}) = 144 $
• $ n ( \text{T}) = 123 $
• $ n ( \text{C}) = 132 $
• $ n ( \text{F} \cap \text{T}) = 58 $
• $ n ( \text{T} \cap \text{C}) = 25 $
• $ n ( \text{F} \cap \text{C}) = 63 $

We know that,

$ n (\text{A} \cup \text{B} \cup \text{C}) = n(\text{A}) + n(\text{B}) + n(\text{C}) – n(\text{A} \cap \text{B}) – n(\text{B} \cap \text{C}) – n(\text{A} \cap \text{C}) + n (\text{A} \cap \text{B} \cap \text{C}) $

Now,

$ n (\text{F} \cup \text{T} \cup \text{C}) = n(\text{F}) + n(\text{T}) + n(\text{C}) – n(\text{F} \cap \text{T}) – n(\text{T} \cap \text{C}) – n(\text{C} \cap \text{F}) + n (\text{F} \cap \text{T} \cap \text{C}) $

$ \Rightarrow 256 = 144 + 123 + 132 – 58 – 25 – 63  + n (\text{F} \cap \text{T} \cap \text{C}) $

$ \Rightarrow n (\text{F} \cap \text{T} \cap \text{C}) = 256 – 253 $

$ \Rightarrow n (\text{F} \cap \text{T} \cap \text{C}) = 3 $

Now, the number of members who can play only tennis $ = n(\text{T})  – n ( \text{F} \cap \text{T}) –  n ( \text{T} \cap \text{C}) – n (\text{F} \cap \text{T} \cap \text{C})$

$\qquad = 123 – 58 – 25 + 3 = 43.$

Correct Answer $: \text{D} $