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?
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?
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.
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