Permutation count of "AEIOUBCDF" where vowels are ascending and consonants are descending

Question

Say we have the string "AEIOUBCDF". How many ways can this string be permuted such that the vowels remain in ascending order, and the consonants remain in descending order (e.g. "AFDEICOUB" is a valid permutation)?

Answer 1

Hint:

You have $9$ slots to fill. Choose $5$ slots for the vowels. Everything else is determined.

Answer 2

A slightly different approach is this:

Consider all 9! possible permutations.
But only one permutation of the vowels is valid -> divide by 5!
and only one permutation of the consonants is valid -> divide by 4!

That gives you 9! / (4! 5!) which is exactly the formula for C(9,5) = C(9,4).