Students liking at least one subject – UGC NET Question

The number of students who like at least one of the two subjects is given by the union of the sets of students liking Mathematics and Science. Using the inclusion–exclusion principle, we have n(M ∪ S) = n(M) + n(S) − n(M ∩ S) = 35 + 30 − 15 = 50. This counts each student only once, even if they like both subjects. Therefore, 50 students like at least one of the two subjects.

Option A:
A value of 45 would incorrectly subtract too many students or miscount those in the intersection. If we tried to get 45 from the given numbers, we would be effectively assuming a different overlap than 15. Since the data provided are precise, 45 does not satisfy the formula for the union. Hence, this option is incorrect.

Option B:
The value 50 is correct because it properly applies the inclusion–exclusion formula for two sets. It ensures that the 15 students who like both subjects are not double counted. In many reasoning questions, this technique is crucial for handling overlapping groups accurately.

Option C:
A value of 55 would imply that n(M ∪ S) is larger than either individual group but fails the inclusion–exclusion calculation 35 + 30 − 15. This would effectively overcount the intersection instead of subtracting it once. Therefore, 55 cannot be the right answer.

Option D:
Taking 60 as the answer would incorrectly assume that every student in the class likes at least one of the two subjects. The union we calculated is only 50, which means 10 students like neither Mathematics nor Science. Thus, 60 is not consistent with the given data.