Question

In a class, there are $200$ students in which $120$ take Mathematics, $90$ take Physics, $60$ take Chemistry, $50$ take Mathematics and Physics, $50$ take Mathematics and Chemistry, $43$ take Physics and Chemistry and $38$ take Mathematics, Physics and Chemistry. Then the number of students who have taken exactly one subject is

Answer 1

Let $M,P,C$ denote the set of students who have taken Mathematics, Physics and Chemistry respectively.
$n\left(M\right)=120,n\left(P\right)=90,n\left(C\right)=60$
$n(M\cap P)=50,n(M\cap C)=50,n(P\cap C)=43,n(M\cap P\cap C)=38$
$\therefore$ Number of students taking exactly one subject
$=n\left(M\right)+n\left(P\right)+n\left(C\right)$
$-2n(M\cap P)-2n(P\cap C)-2n(M\cap C)+3n(M\cap P\cap C)$
$=120+90+60-2\left(50\right)-2\left(43\right)-2\left(50\right)+3\left(38\right)$
$=98$