In a class of 140 students numbered 1 to 140, all even numbered students opted mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is?

Let n(A) = number of students opted Mathematics = 70, n(B) = number of students opted Physics = 46, n(C) = number of students opted Chemistry = 28, n(A∩B) = 23, n(B∩C) = 9, n(A∩C) = 14, n(A∩B∩C) = 4.

Now n(A∪B∪C) = n(A) + n(B) + n(C) - n(A∩B) - n(B∩C) - n(A∩C) + n(A∩B∩C) = 70 + 46 + 28 - 23 - 9 - 14 + 4 = 102

So number of students not opted for any course = Total - n(A∪B∪C) = 140 - 102 = 38.