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. The number of student who study all the three subject is

• A. $30$
• B. $20$
• C. $22$
• D. $25$

Answer 1

Answer: (B)

$20$

Let M, P and C denote the students studying Mathematics, Physics and Chemistry

And $U$ represents total students

So, $n\left(U\right)=200$,

$n\left(M\right)=120,n\left(P\right)=90$

$n\left(C\right)=70,n(M\cap P)=40,n(P\cap C)=30$

$n(M\cap C)=50,n(M\cup P\cup C{)}^{\prime }=20$

$\therefore n(M\cup P\cup C{)}^{\prime }=n(U)-n(M\cup P\cup C)$

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

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

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

$-n(M\cap P)-n(P\cap C)-n(C\cap M)+n(C\cap M\cap P)$

$\therefore 180=120+90+70-40-30-50+n(C\cap M\cap P)$

$\Rightarrow 180=280-120+n(C\cap M\cap P)$

$\Rightarrow n(P\cap C\cap M)=300-280=20$

Hence, the number of students studying all three subjects is $20$.