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

In how many different ways can the letters of the word 'DETAIL' be arranged in such away that the vowels occupy only the odd positions? Correct Answer 36

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.