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.

Out of 100 students, 20% failed in both subjects, so 80 students passed in at least one of Mathematics or English. Let the number who passed in both subjects be x. Using inclusion–exclusion, the number who passed in at least one is 60 + 70 − x. This must equal 80, giving 130 − x = 80 and hence x = 50. Therefore, 50 students passed in both Mathematics and English.

Option A:
Option A, 40, would correspond to 60 + 70 − 40 = 90 students passing in at least one subject, which contradicts the given figure that only 80 passed at least one.

Option B:
Option B, 45, gives 60 + 70 − 45 = 85 students passing in at least one, again inconsistent with the total implied by 20 students failing both.

Option C:
Option C, 55, implies 60 + 70 − 55 = 75, which would mean 25 students failed in both subjects, not the stated 20.

Option D:
Option D, 50, exactly balances the inclusion–exclusion equation and yields 80 students passing at least one subject, matching the information derived from the 20 who failed both.