UGCNET Questions

The word LEVEL has five letters in total, with L appearing twice and E appearing twice, while V appears once. The formula for permutations of n objects with repetitions is n factorial divided by the product of factorials of repeated counts. Thus, the number of distinct permutations is 5! ÷ (2! × 2!). Calculating this gives 120 ÷ 4 = 30. Hence, there are 30 different arrangements of the letters in LEVEL.

Option A:
Option A, 20, underestimates the actual number and cannot be obtained by the correct factorial expression 5! ÷ (2! × 2!). It likely comes from an incorrect guess or misapplied formula.

Option B:
Option B accurately uses the repetition-adjusted permutation formula, recognizing two repeated L’s and two repeated E’s. By evaluating 120 ÷ 4, it yields 30, which counts all unique arrangements without overcounting identical ones.

Option C:
Option C, 40, exceeds the true count and would require a different numerator or denominator in the formula. It suggests that the effect of repeated letters has not been properly accounted for.

Option D:
Option D, 60, is exactly half of 120 and may arise from dividing by only one 2! instead of both 2!. This partially correct adjustment still leads to overcounting because both L and E are repeated.

The word NET has three distinct letters: N, E and T. The number of permutations of n distinct objects taken all at a time is n!, so here it is 3!. Calculating 3! gives 3 × 2 × 1 = 6. Therefore, there are 6 possible distinct arrangements of the letters in NET.

Option A:
Option A uses the factorial formula for permutations correctly, recognizing that there are no repeated letters. By computing 3! as 6, it counts all possible orderings such as NET, NTE, ENT, ETN, TEN and TNE.

Option B:
Option B, 3, would be the number of letters, not the number of arrangements. It undercounts the permutations by ignoring the different orders in which letters can appear.

Option C:
Option C, 9, may come from incorrectly squaring the number of letters (3²), which does not match the combinatorial rule for arrangements of distinct objects.

Option D:
Option D, 12, overestimates the possible arrangements and could arise from misapplying permutation formulas that involve repetition or additional positions.