Question

In a group of $800$ people, $500$ can speak Hindi and $320$ can speak English. Let $\alpha$ be the number of people who speak both languages. Then

• A. $320\le \alpha \le 500$
• B. $20\le \alpha \le 320$
• C. $\alpha \le 320$
• D. $\alpha \ge 320$

Answer 1

The correct option is B $20\le \alpha \le 320$

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

$n\left(U\right)=800,n\left(H\right)=500,n\left(E\right)=320$
$n(H\cap E)=\alpha$
$n(H\cup E)=n\left(H\right)+n\left(E\right)-n(H\cap E)$
$\Rightarrow n(H\cup E)=820-\alpha$
$\Rightarrow \alpha =820-n(H\cup E)$

For $\alpha$ to be minimum, $n(H\cup E)$ should be maximum.
${\alpha }_{min}=820-800=20$

Also, $\alpha \le min\left\{n\right(H),n(E\left)\right\}=320$
Hence, $20\le \alpha \le 320$