UGCNET Questions

To find how many students like at least one of the two subjects, we use the inclusion–exclusion principle. We add those who like each subject and subtract those counted twice in the overlap: 22 + 18 − 10 = 30 students. These 30 students like Mathematics, Science or both.

Option A:
Option A, 30, is the direct result of applying inclusion–exclusion and correctly represents the number who like at least one of the two subjects.

Option B:
Option B, 32, would correspond to assuming an overlap of 8 instead of 10, which contradicts the given overlap and misuses the data.

Option C:
Option C, 40, assumes that every student likes at least one of the subjects, which is not stated and need not be true.

Option D:
Option D, 50, exceeds the total number of students in the group, so it is impossible.

Using the principle of inclusion–exclusion, the number of students who like at least one of Mathematics or English is 18 + 16 − 8 = 26. The total number of students in the class is 30. Therefore, those who like neither subject are 30 − 26 = 4. This calculation separates students who fall outside both preference sets.

Option A:
Option A, 2, would imply that 28 students like at least one of the two subjects, but the inclusion–exclusion result clearly shows only 26 students in that category. Hence, 2 underestimates the count of those liking neither.

Option B:
Option B, 3, similarly assumes 27 students like at least one subject, conflicting with the properly derived figure of 26. It reflects a minor miscalculation in subtracting from the total.

Option C:
Option C correctly applies inclusion–exclusion to avoid double counting those who like both subjects, then subtracts from the total class strength. This yields 4 students who like neither Mathematics nor English.

Option D:
Option D, 6, would require at least one count to be different because it implies only 24 students like at least one subject, which contradicts the overlap-adjusted total of 26.

Let the total number of people be represented as 100%. If 20% like neither tea nor coffee, then 80% like at least one of the two beverages. By the inclusion–exclusion principle, the percentage who like at least one is the sum of tea-likers and coffee-likers minus those who like both. Thus, 80 = 70 + 40 − (both). Solving this gives (both) = 70 + 40 − 80 = 30%.

Option A:
Option A, 20%, equals the proportion who like neither, not those who like both, and so confuses the outside region with the intersection.

Option B:
Option B calculates the intersection using inclusion–exclusion and ensures that the total does not exceed 100%. It matches the only value that makes the percentages internally consistent.

Option C:
Option C, 40%, would give 70 + 40 − 40 = 70% liking at least one, which contradicts the information that 20% like neither and thus only 80% like at least one.

Option D:
Option D, 50%, creates an even larger overlap and leads to a sum exceeding the allowed total, making the data inconsistent.