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Q.
In a group of $800$ people, $550$ can speak Hindi and $450$ can speak English. How many can speak both Hindi and English?
Sets
Solution:
Let $H$ denote the set of people speaking Hindi and $E$ denote the set of people speaking English. We are given that $n(H) = 550$, $n(E) = 450$ and $n(H \cup E ) = 800$
We know that $n(H \cup E) = n(H) + n(E) - n(H \cap E)$
$\Rightarrow n(H \cap E) = n(H) + n(E) - n(H \cup E)$
$\Rightarrow n(H \cap E) = 550 + 450 - 800 = 200$
Hence, $200$ persons can speak both Hindi and English.