There are then 'slots' into which arrangements of vowels can be placed. We must not place the 'e' alone in a slot, or there would then be one or more consonants between the 'e' and another vowel. So, we'll assume first that the 'e' is placed with another single vowel, and the remaining vowel in a separate slot.
There are ways of placing the 'e' into a slot, and then 4 ways of choosing and positioning an 'a' or an 'o' in the same slot alongside the 'e'. There are then ways of placing the third vowel into a different slot. Total: . But in half of these arrangements the third vowel will be placed in a slot where there is a group of consonants between the 'e' and this third vowel. So the permitted number of ways of positioning the group of 2 vowels and a single third vowel is .
Then there are ways of arranging the 3 vowels into a single group, and ways of choosing a 'slot' for this group. Total: .
There are therefore ways of positioning the vowels into the 'slots' formed by the consonants.
So the total number of arrangements is .