Question

In a group of $400$ people, $250$ can speak Hindi and $200$ can speak English. How many people can speak both Hindi and English?

Answer 1

Let H be the set of people who speak Hindi, and E be the set of people who speak English
$\therefore n(H\cup E)=400,n\left(H\right)=250,n\left(E\right)=200$
$n(H\cap E)=?$

We know that:
$n(H\cup E)=n\left(H\right)+n\left(E\right)-n(H\cap E)$
$\therefore 400=250+200-n(H\cap E)$
$\Rightarrow 400=450-n(H\cap E)$
$\Rightarrow n(H\cap E)=450-400$
$\therefore n(H\cap E)=50$

Thus, 50 people can speak both Hindi and English.