Question

In a survey of $200$ students of a higher secondary school, it was found that $120$ studied mathematics; $90$ studied physics; and $70$ studied chemistry; $40$ studied mathematics and physics; $30$ studied physics and chemistry ; $50$ studied chemistry and mathematics, and $20$ studied none of these subjects. Find the number of students who studied all the three subjects.

Answer 1

Given $n\left(U\right)=200,n\left(M\right)=120\left(p\right)=90,$ $n\left(C\right)=70;n(M\cap P)=40,n(P\cap C)=30$ $n(C\cap M)=50,n(M^{\prime }\cap P^{\prime }\cap C^{\prime })=20$
Since, $n(M^{\prime }\cap P^{\prime }\cap C^{\prime })=20$
$\Rightarrow n{(M\cup P\cup C)}^{\prime }=20$ (by Demorgan's law)
$\Rightarrow n\left(U\right)-n(M\cup P\cup C)=20$
$\Rightarrow n(M\cup P\cup C)=180.$
$\Rightarrow n\left(M\right)+n\left(P\right)+n\left(C\right)-n(M\cap P)-n(P\cap C)-n(C\cap M)+n(M\cap p\cap C)$
$\therefore 180=(120+90+70)-(40+30+50)+n(M\cap P\cap C)$
$\Rightarrow n(M\cap P\cap C)=20.$
Hence 20 students studied all the three subjects.