If the letters of the word 'daughter ' are to be arranged so that vowels occupy the odd places , then the number..

There are only 3 vowels, a, u, and e, and there are 4 odd places.  So
one of the odd spaces must contain a consonant.

We can choose 3 of the 4 odd places to put the vowels in C(4,3) or 4 ways.

For each of those vowel , we can  rearrange them  in P(3,3) or 3! or 6 ways.

And we can rearrange the 5 consonants in P(5,5) or 5! or 120 ways.

Hence the total number of different words formed are 6*4*120 = 2880