Out of 80 students who appeared for the school exams in Mathematics (M), Physics (P) and Chemistry (C), 50 passed M, 30 passed P and 40 passed C. At most 20 students passed M and P, at most 20 students passed P and C and at most 20 students passed C and M. The maximum number of students who could have passed all three exams is _____________.

Let the number of students who passed in all three subjects be x.

$$M∪P∪C=M+P+C-\left(M∩P\right)-\left(M∩C\right)-\left(P∩C\right)+\left(M∩P∩C\right)$$

$$80=50+30+40-\left(M∩P\right)-\left(M∩C\right)-\left(P∩C\right)+x$$

$$80=120-\left(M∩P\right)-\left(M∩C\right)-\left(P∩C\right)+x$$

$$\left(M∩P\right)+\left(M∩C\right)+\left(P∩C\right)-40=x$$

The value of x will be maximum when maths and physics, physics and chemistry, and maths and chemistry values are maximum, which is equal to 20.

$$20+20+20-40=x$$

=> $$x=20$$