Question

If in a class of $100$ students, $60$ like mathematics, $72$ like physics, $68$ like chemistry and no student likes all three subjects, then the number of students who don't like mathematics and chemistry is

Answer 1

Let $M$ be the set of people who like Mathematics;
$P$ is the set of people who like Physics;
$C$ is the set of people who like chemistry
$\because$ No student like all three subjects, the situation can be represented as:

$100=60+72+68-(x+y+z)+0\Rightarrow x+y+z=100$

So no student like only maths, or only physics or only chemistry

$\begin{array}{c}x+y=60\\ x+z=72\\ y+z=68\end{array}\}\Rightarrow y=28$

Now,
$n(M^{\prime }\cap C^{\prime })=100-n(M\cup C)$
$=100-(60+68-y)=28-28=0$