Question

In a survey of population of 450 people, it is found that 205 can speak English (E), 210 can speak Hindi (H) and 120 people can speak Tamil, If 100 people can speak both E and H, 80 can speak both E and T, 5 can speak both H and T, and 20 can speak all the three languages, find the number of people who can speak E but not H or T. Find also the number of people who can speak neither E nor H nor T.

Answer 1

$n\left(U\right)=450,n\left(E\right)=205,n\left(H\right)=210,n\left(T\right)=120$
$n(E\cap H)=100,n(E\cap T)=80,n(H\cap T)=35$
$n(E\cap H\cap T)=20.$
Number of people who can speak E but not H and T is
$n(E\cap H^{\prime }\cap T^{\prime })=n(E\cap {(H\cup T)}^{\prime })$
$=n\left(E\right)-n(E\cap (H\cup T))$
$=n\left(E\right)-n\{(E\cap H)\cup (E\cap T)\}$
$=n\left(E\right)-[n(E\cap H)+n(E\cap T)-n\{(E\cap H)\cap (E\cap T)\}]$
$=n\left(E\right)-n(E\cap H)-n(E\cap T)+n(E\cap H\cap T)$
$=205-100-80+20$
$\Rightarrow n(E\cap H^{\prime }\cap T^{\prime })=225-180=45$
Next, we have to find number of people who cannot speak E,H and T
$n(E^{\prime }\cap H^{\prime }\cap T^{\prime })=n{(E\cup H\cup T)}^{\prime }$
$=n\left(U\right)-n(E\cup H\cup T)$ ....(1)
$n(E\cup H\cup T)=n\left(E\right)+n\left(H\right)+n\left(T\right)-n(E\cap H)-n(H\cap T)-n(E\cap T)+n(E\cap H\cap T)$
$\Rightarrow n(E\cup H\cup T)=(205+210+120)-(100+80+35)+20=340$
Hence from (1)
$n(E^{\prime }\cap H^{\prime }\cap T^{\prime })=450-340=110$