# How many words can be formed from the letters of the word 'DAUGHTER' so that the vowels never come together?

How many words can be formed from the letters of the word 'DAUGHTER' so that the vowels never come together ?

The answer is obviously $8!-6!\cdot3!$.

My question is that if we ponder from a different perspective, that is taking $5$ consonants first and arranging them ($5!$ ways of doing that) and then placing the $3$ vowels in the $6$ places created due to the arrangement of consonants ($\frac{6!}{3!}$ ways to do that), the answer should be $5!\frac{6!}{3!}$.

What is wrong with this?

-

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"?