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

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Let $H$ be the set of people who speak Hindi, and E be the set of people who speak English

$\therefore n(H \cup E)=400, n(H)=250, n(E)=200$

$n(H \cap E)=?$

We know that:

$n(H \cup E)=n(H)+n( E )-n(H \cap E)$

$\therefore 400=250+200-n(H \cap E)$

$\Rightarrow 400=450-n(H \cap E)$

$\Rightarrow n(H \cap E)=450-400$

$\therefore n(H \cap E)=50$

Thus, $50$ people can speak both Hindi and English.

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