Question

The report of one survey of $100$ student stated that the numbers studying the various language were: Sanskrit, Hindi and Tamil $5$; Hindi and Sanskrit $10$; Tamil and Sanskrit $8$; Hindi and Tamil $20$; Sanskrit $30$; Hindi $23$; Tamil $50$. Find the number of students who took none of the three languages.

Answer 1

Given $n\left(S\right)=30,n\left(H\right)=23,n\left(T\right)=50,n(H\cap T)=20,n(H\cap S)=10,n(T\cap S)=8,$

$n(H\cap S\cap T)=5,n\left(U\right)=100$
We have
$n(H\cup S\cup T)=n\left(H\right)+n\left(S\right)+n\left(T\right)-n(H\cap T)-n(H\cap S)-n(S\cap T)+n(H\cap S\cap T)$
$\Rightarrow n(H\cup S\cup T)=103-38+5=70$

Now, the number of students who took none of three langugaes

$=n(H\cup S\cup T{)}^{\prime }$

$=n\left(U\right)-n(H\cup S\cup T)$

$=100-70$

$=30$

So, $30$ students did not take any of the three languages.