In a class of $150$ students, $70$ students passed in Mathematics, $60$ students passed in Chemistry and $20$ students passed in both Mathematics and Chemistry. Then the number of students that failed in both the subjects are:

Let the total number of students be $n(U)=150$.
* Let students who passed in Mathematics be $n(M)=70$.
* Let students who passed in Chemistry be $n(C)=60$.
* Let students who passed in both subjects be $n(M\cap C)=20$.

We find the total number of unique students who passed at least one subject using the set union formula:

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

$$n(M\cup C)=70+60-20=110$$

The number of students who failed in both subjects is the complement of the union:

$$\text{Failed Students}=n(U)-n(M\cup C)=150-110=40$$