Question

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?

Answer 1

Let C denote the set the people like cricket, and T denote the set of people who like tennis
$\therefore n(C\cup T)=65,n\left(C\right)=40,n(C\cap T)=10$

We know that
$n(C\cup T)=n\left(C\right)+n\left(T\right)-n(C\cap T)$
$\therefore 65=40+n\left(T\right)-10$
$\Rightarrow 65=30+n\left(T\right)$
$\Rightarrow n\left(T\right)=65-30=35$

Therefore, 35 people like tennis.

Now,
$n(T-C)=n\left(T\right)-n(T\cap C)$
$\Rightarrow n(T-C)=35-10=25$

Thus, 25 people like only tennis.