There are 6 letters in the given word, out of which there are 3 vowels and 3 consonants.

Let us mark these positions as under:

(1) (2) (3) (4) (5) (6)

Now, 3 vowels can be placed at any of the three places out 4, marked 1, 3, 5.

Number of ways of arranging the vowels = ³P₃ = 3! = 6.

Also, the 3 consonants can be arranged at the remaining 3 positions.

Number of ways of these arrangements = ³P₃ = 3! = 6.

Total number of ways = (6 x 6) = 36.

To arrange the vowels (EAI) in the odd positions, we have 3 choices for the first odd position (either E or A or I) and 2 choices for the second odd position (one of the two remaining vowels).

Thus, we have 3 × 2 = 6 ways to arrange the vowels in the odd positions.

Now, we need to fill the even positions with the remaining consonants (D, T, and L). The consonants can be arranged in the remaining 3 even positions in 3! ways.

Therefore, the total number of ways to arrange the letters of the word DETAIL such that the vowels occupy only the odd positions is 6 × 3! = 36.

Hence, the correct answer is option C (36).