UGC NET Questions (Paper – 1)

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.

Statements A, B, D and E are all correct descriptions of the role of permutations and combinations. Permutations count ordered arrangements, combinations count unordered selections, seating arrangement problems rely heavily on order-based counting, and both concepts are frequently tested in UGC NET aptitude. Statement C is false because, for r greater than 1, nPr is larger than nCr since each combination can be arranged in several ordered ways. Thus the only option that includes all true statements while excluding the false one is the combination A, B, D and E.

Option A:
Option A is incomplete because it omits E, failing to note explicitly that both permutations and combinations are actively used in NET mathematical reasoning questions. Without E, the statement about exam relevance is missing.

Option B:
Option B is correct because it gathers A, B, D and E, which together present the conceptual distinction between order and selection and their examination usage, while rightly rejecting C, which reverses the relative sizes of permutations and combinations.

Option C:
Option C is incorrect since it leaves out A and therefore does not mention that permutations are arrangements where order matters, a central concept for the topic. Though it contains some true statements, it is not complete.

Option D:
Option D is wrong because it includes C, the false claim that nPr is always less than nCr, and thereby mixes an incorrect statement with true ones. The presence of C makes the entire combination unacceptable.

A and B correctly give the formulas for permutations and combinations without repetition. C is true because allowing repetition means each of the r positions can be filled in n ways, giving nʳ arrangements. E is also correct since identifying whether order matters is essential to selecting the right counting approach. D is false because without repetition, r-length arrangements from n objects are counted by nPr, not rⁿ. Thus A, B, C and E only are correct.

Option A:
Option A misses E and therefore does not state the important strategic step of checking whether order is significant. Although it contains correct formulas, it is not complete as an answer.

Option B:
Option B omits A and so fails to mention the basic permutation formula, even though it includes other true statements. Leaving out A makes the set incomplete for the topic.

Option C:
Option C is correct because it collects all the true statements while excluding D, which misapplies exponent notation. It reflects the range of arrangements encountered in NET questions.

Option D:
Option D excludes B and so ignores the combination formula, and it does not correct the error in D. Consequently, it does not represent the full correct set.