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.  

Take an ordering of the consonants (there are $21!$ such orderings), for example:
$$bcdfghjklmnpqrstvwxyz$$
Enumerate the 22 "slots" between these consonants:
$$\_b\_c\_d\_f\_g\_h\_j\_k\_l\_m\_n\_p\_q\_r\_s\_t\_v\_w\_x\_y\_z\_$$
We can choose (respecting order) $_{22} P _5$ lists of five slots to put the vowels $aeiou$ in order.
The answer is therefore $_{22} P _5 \cdot 21!$
