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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?

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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"?

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