How many words can be formed from the letters of the word 'DAUGHTER' so that the vowels never come together?
Solution 1:
Your first solution ($36000$) counts words in which AEU don't come all three together, your second solution ($14400$) counts those in which they are all three separate, which is quite a more strict condition. Which shows that it helps to pose your question more carefully: what exactly do you mean by "the vowels come together"?