Question

In a survey of $200$ students of a higher secondary school it was found that $120$ studied Mathematics (M), $90$ studied Physics (P) and $70$ studied Chemistry (C), $40$ studied M and P, $30$ studied P and C, $50$ studied C and M, and $20$ studied none of these subjects. Find the number of students who studied all the three subjects.

Answer 1

Given $n\left(U\right)=200,n\left(M\right)=120,n\left(P\right)=90,n\left(C\right)=70$
$n(M\cap P)=40,n(P\cap C)=30,n(C\cap M)=50$
$n{(M\cup P\cup C)}^{\prime }=20$
$\Rightarrow n\left(U\right)-n(M\cup P\cup C)=20$
$\Rightarrow n(M\cup P\cup C)=180$
$\Rightarrow n\left(M\right)+n\left(P\right)+n\left(C\right)-[n(M\cap P)+n(P\cap C)+n(C\cap M)]+n(M\cap P\cap C)=180$
or $(120+90+70)-[40+30+50]+n(M\cap P\cap C)=180$
$\therefore n(M\cap P\cap C)=180-160=20$
Hence 20 studied all the three subjects.