Question

In a group of 1000 people , there are 750 who can speak Hindi and 400 who can speak Bengali. How many can speak Bengali ?

Answer 1

Let H and B denote the sets of those people who can speak Hindi and Bengali respectively. Given $n(H\cup B)=1000,n\left(H\right)=750,n\left(B\right)=400.$
Now $n(H\cup B)=n\left(H\right)+n\left(B\right)-n(H\cap B)$
$1000=750+400-n(H\cap B)$
$\Rightarrow (H\cap B)=150.$
Hence 150 people can speak both Hindi and Bengali.
Number of people who can speak Bengali only
$n(B\cap H^{\prime })=n\left(B\right)-n(B\cap H)$
$=400-150=250$
Hence, 250 people can speak Bengali only.