The letters A, B, C, D and E are arranged in such a way that there are exactly two letters between A and E. How many such arrangements are possible?

(a) 12
(b) 18
(c) 24 ✓ UPSC's answer
(d) 36

Why the answer is (c)

• The condition 'exactly two letters between A and E' means A and E must occupy positions (1,4) or (2,5) in a 5-letter sequence.

• There are 2 possible position pairs for A and E: (1,4) and (2,5).

• For each position pair, A and E can be arranged in 2 ways (A at the first position, E at the second, or vice versa).

• The remaining 3 letters (B, C, D) can be arranged in the remaining 3 positions in 3! = 6 ways.

• Total arrangements = 2 (position pairs) × 2 (A/E order) × 6 (B/C/D order) = 24.

• Therefore, the correct number of arrangements is 24.

Why the other options are wrong

(a) 12: 12 is incorrect because it likely accounts for only one position pair or fails to multiply by the permutations of the remaining three letters correctly.
(b) 18: 18 is incorrect because it does not correspond to any valid combinatorial calculation for this specific constraint.
(d) 36: 36 is incorrect because it overcounts the possibilities, possibly by assuming more position pairs or incorrect permutations for the remaining letters.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.