Question

In a survey of $100$ students the number of students studying various languages is found as : English only $18$, English but not Hindi $23$; English and Sanskrit $8$; Sanskrit and Hindi $8$, English $26$, Sanskrit $48$ and no language $24$. Find
(1) How many students are studying Hindi?
(2) How many students are studying both English and Hindi?

Answer 1

Let $\cap$ denotes the set of surveyed students.

Let E,H and S denotes the set of students who are studying English, Hindi and Sanskrit respectively.

Then, $n\left(U\right)=100$

$n\left(E\right)=26$

$n\left(S\right)=48$

$n(E\cap S)=8$

$n(S\cap H)=8$

From the Venn-diagram, it is clear that

$n(E\cap H\cap S)=3$

The number of students who study English only $=18$

Number of students who study no language $=24$

$\therefore$ Number of students who study Hindi only $=[100-(18+5+3+5+35\left)\right]-24$

$=100-66-24$

$=100-90$

$=10$

$\therefore$ Number of students who study Hindi $=10+3+5$

$=18$

And Number of students who study English and Hindi $=3$