Question

In a survey of $200$ students of a school, it was found that $120$ study Mathematics, $90$ study Physics and $70$ study Chemistry, $40$ study Mathematics and Physics, $30$ study Physics and Chemistry, $50$ study Chemistry and Mathematics and $20$ none of these subjects. Find the number no. of students who study all the subject.

Answer 1

Total students $200{=}_n\left(U\right)$

No. of students with Mathematics $120{=}_n\left(M\right)$

No. of students with physics $=90{=}_n\left(P\right)$

No. of students with chemistry $=70{=}_n\left(C\right)$

No. of students with Maths and Physics $=40{=}_n(M\cup P)$

No. of students with Physics and Chemistry $=30{=}_n(P\cup C)$

No. of students with Chemistry and Maths $=50{=}_n(C\cup M)$

No. of students who study none $=20{=}_n\left(U\right){-}_n(M\cup P\cup C)$

$\Rightarrow n(M\cup P\cup C)=200-20=180$

Now,

$n(M\cap P\cap C)=n(M\cup P\cup C)+n(M\cap C)+n(M\cap C)+n(P\cap C)-n\left(M\right)-n\left(P\right)-n\left(C\right)$

$180+40+30+50-70-90-120$

$=20$