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

DETAIL is a six-letter word with three vowels and three consonants. 
This means there will be three odd places and three even places. 
This means we must fit the three vowels in the three odd places. And the three consonants can fit into the three even places. 
The number of ways to arrange three items in three places will be 3!. 

We can arrange three vowels in three odd places in 3! ways, and three consonants in three even places in 3! ways. 

Total ways to arrange it will be (3!)(3!)=36.