Number of words having vowels in alphabetical order, taken from the word EDUCATION

Question

The question goes as follows:

If words are formed using letters of the word 'EDUCATION', then how many words will have vowels in alphabetical order?

I tried making cases but the cases became very extensive, I started getting many cases since there were 4 consonants and 5 vowels. Any help would be appreciated.

Answer 1

EDUCATION

9 Letters(5 vowels & 4 consonants)

There are 9! ways to arrange letters of 'EDUCATION'

Lets just forget about consonants and focus on vowels. We have E,U,A,I,O vowels present.

Now we can arrange these vowels in 5! ways and you will find out that there is only 1 out of 120 ways where all vowels are in alphabetical order i.e.(A-E-I-O-U).

Therefore our answer is $\frac{9!}{5!}$