105 students take an examination of whom 80 students pass in English. 75 students pass in Mathematics and 60 students pass in both subjects. How many students fail in both subjects?

A simple application of inclusion-Exclusion Principle.
Let X= the set of students who take the examination.
A = the set O students who pass in English
B = the set of students who pass in Mathematics
We are given that  $n\left(x\right)=105,n\left(A\right)=80,n\left(B\right)=75,n(A\cap B)=60$
Since,  $n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cup B).$
Therefore,  $n(A\cup B)=80+75-60=95.$
The required number  $n\left(x\right)-n(A\cup B)=105-95=10$
 Thus, 10 students fail in both subjects.