In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only

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1.Set Theory

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In a group of $65$ people, $40$ like cricket, $10$ like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?

A

$25$

B

$25$

C

$25$

D

$25$

Solution

Let $C$ denote the set of people who like cricket, and $T$ denote the set of people who like tennis

$\therefore n(C \cup T)=65, n(C)=40, n(C \cap T)=10$

We know that:

$n(C \cup T)=n(C)+n(T)-n(C \cap T)$

$\therefore 65=40+n(T)-10$

$\Rightarrow 65=30+n(T)$

$\Rightarrow n(T)=65-30=35$

Therefore, $35$ people like tennis.

Now,

$(T-C) \cup(T \cap C)=T$

Also.

$(T-C) \cap(T \cap C)=\varnothing$

$\therefore n(T)=n(T-C)+n(T \cap C)$

$\Rightarrow 35=n(T-C)+10 $

$\Rightarrow n(T-C)=35-10=25$

Thus, $25$ people like only tennis.

Standard 11

Mathematics

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