Correct Option (C)
The problem requires arranging five distinct letters (A, B, C, D, E) such that exactly two letters are positioned between A and E.
This can be solved in two steps:
- Determine the possible arrangements for the pair (A, E) satisfying the condition.
- Arrange the remaining three letters (B, C, D) in the vacant positions.
For step 1, consider the five available positions (P1, P2, P3, P4, P5):
- If A is at P1, E must be at P4 (A _ _ E _).
- If A is at P2, E must be at P5 (_ A _ _ E).
- If E is at P1, A must be at P4 (E _ _ A _).
- If E is at P2, A must be at P5 (_ E _ _ A).
Thus, there are 4 distinct positional arrangements for the pair (A, E).
For step 2, after placing A and E, 3 positions remain vacant. The remaining 3 letters (B, C, D) can be arranged in these 3 positions in 3! ways.
3! = 3 × 2 × 1 = 6 ways.
The total number of such arrangements is the product of the possibilities from both steps:
Total arrangements = (Number of A-E arrangements) × (Arrangements of B, C, D)
Total arrangements = 4 × 6 = 24.
Incorrect Options:
(A) 12: This result would be obtained if only one order of A and E (e.g., A before E) was considered, leading to 2 arrangements for A and E, multiplied by 3! (2 × 6 = 12). It fails to account for the cases where E precedes A.
(B) 18: This value does not align with a standard permutation or combination calculation for this problem. It might arise from an incorrect count of A-E placements or an error in calculating permutations of the remaining letters.
(D) 36: This result could occur if the number of possible A-E arrangements was incorrectly calculated as 6 instead of 4, and then multiplied by 3! (6 × 6 = 36). An incorrect count of the initial A-E placements leads to this overestimation.