Question

The report of one survey of $100$ students stated that the numbers studying the various languages were : Sanskrit, Hindi and Tamil, $5$; Hindi and Sanskrit, $10$; Tamil and Sanskrit, $8$; Hindi and Tamil, $20$; Sanskrit $30$; Hindi $23$; Tamil $50$. The surveyor who prepared the report was fired. Why ?

Answer 1

Let $S,H$ and $T$ denote the set of students studying Sanskrit, Hindi and Tamil respectively. Then we are given :
$n(S\cup H\cup T)=100,n\left(S\right)=30,n\left(H\right)=23,$
$n\left(T\right)=50,n(S\cap H)=10,n(H\cap T)=5.$
Now $n(S\cup H\cup T)=n\left(S\right)+n\left(H\right)+n\left(T\right)-n(S\cap H)-n(H\cap T)-n(T\cap S)+n(S\cap H\cap T)$
$=S_1-S_2+S_3$
$=(30+23+50)-(10+20+8)+5=70.$
But we are given $n(S\cup H\cup T)=100.$
Hence the data is inconsistent. That is why the surveyor was fired.