Question

Out of the $200$ residents of any neighbourhood, $120$ can speak Hindi, $60$ can speak English and $50$ can speak Tamil. $10$ residents can speak both Hindi and English, $15$ residents can speak English and Tamil, while $15$ residents can speak both Hindi and Tamil. $10$ residents can speak all three languages. Find number of residents who can speak Hindi but not Tamil.

• A. $105$
• B. $135$
• C. $90$
• D. None of the above

Answer 1

The correct option is A $105$

Let $H$ be the set of all Hindi speaking residents,
$E$ be the set of all English speaking residents,
$T$ be the set of all Tamil speaking residents,

Given: $n\left(U\right)=200,n\left(H\right)=120,n\left(E\right)=60,n\left(T\right)=50$
$n(H\cap E)=10,n(E\cap T)=15,n(H\cap T)=15,n(H\cap E\cap T)=10$

To find: $n(H-T)$

We know the difference formula,
$n(A-B)=n\left(A\right)-n(A\cap B)$
$\Rightarrow n(T-E)=n\left(T\right)-n(T\cap E)$
$\Rightarrow n(T-E)=120-15=105$