Arrangement permutations with positional constraints between two letters

We have 5 slots: `_ _ _ _ _`. To have exactly two letters between A and E, their positions are fixed to specific structural pairs. The possible index placements (1-based) for the pair (A, E) are:

Since A and E can also swap places (E before A), this gives 2 × 2 = 4 fixed layout patterns for the boundary letters.

For each of these 4 patterns, the remaining 3 empty slots must be filled by the remaining 3 letters (B, C, D). Number of ways to arrange them = 3! = 6 ways.

Total possible arrangements = ⟨MATH⟩4 patterns × 6 arrangements = 24⟨/MATH⟩.

Answer: (c).