In how many different ways can the letters of the word 'DETAIL' be arranged so that the vowels occupy only the odd positions?

Problem restatementPlace the vowels of 'DETAIL' in odd positions (1, 3, 5) and the consonants in the remaining even positions (2, 4, 6).

Given data

• Word: D, E, T, A, I, L (6 distinct letters).
• Vowels: E, A, I (3 vowels).
• Odd positions available: 1, 3, 5 (exactly 3 slots).

Concept/ApproachAssign the 3 vowels to the 3 odd slots and the 3 consonants (D, T, L) to the 3 even slots independently.

Step-by-step calculation Arrange vowels in odd slots: 3! = 6 Arrange consonants in even slots: 3! = 6 Total arrangements = 6 × 6 = 36

Verification/AlternativeBecause the counts match exactly (3 odd slots and 3 vowels), no combinations factor is needed; only permutations within positions.

Common pitfalls

• Allowing vowels in even positions or forgetting to permute consonants separately.

Final Answer36