Question

In a class of $100$ students, $60$ like mathematics, $72$ like physics, $68$ like chemistry and no student likes all three subjects. Then number of students who don't like mathematics and chemistry is

Answer 1

Let M, P and C represents the set of students who likes Mathematics, Physics and Chemistry respectively as shown in venn diagram below:
From the venn diagram, we have $n(M\cap P)=x,n(M\cap C)=y\text{and}n(P\cap C)=z$

As we know,
$n(M\cup P\cup C)=n\left(M\right)+n\left(P\right)+n\left(C\right)-n(M\cap P)-n(M\cap C)-n(P\cap C)+n(M\cap P\cap C)$
$\Rightarrow 100=60+72+68-(x+y+z)+0x+y+z=100$
Since, no student like only maths, or only physics or only chemistry
So $\begin{array}{c}x+y=60\\ x+z=72\\ y+z=68\end{array}\}\Rightarrow y=28$

Now the number of students who don't like mathematics and chemistry is given by
$n(M^{\prime }\cap C^{\prime })=100-n(M\cup C)$
$=100-(60+68-y)=y-28=0$