The number of ways to order 26 alphabet letters, no two vowels occurring consecutively What is the shortest solution to the following problem?

What is the number of ways to order the 26 letters of alphabet so that
  no two of the vowels a,e,i,o,u occur consecutively?

What I thought is to subtract permutations consisting of 2 vowels, 3 vowels, 4 vowels and 5 vowels occurring consecutively from all permutations 26!. But I could not even find an reasonable answer with that method. It seems to be too long and I always make mistakes.  

Separate it into two cases:
Case 1: No vowel occurs at the end of the list. Then all permutation satifying your condition will be the ones where all the vowels have a consonant to the right of it. Then find the number of ways to pair each vowel with a consonant, then treat the pairs as single objects and multiply by the number of permutations of the remaining objects.
Case 2 is solved similarly except you need to discount one of the vowels and multiply by 5 for your choice of vowel to appear on the end
