How many words can be formed using all the letters of "DAUGHTER" so that vowels always come together? How many words can be formed using all the letters of "DAUGHTER" so that vowels always come together?
I understood that there are 6 letters if we consider "AUE" as a single letter and answer would be 6!. Again for AUE it is 3!, but I didn't get why to do 6! * 3!.
Can't we just add (6! + 3!) to get final result?

Imagine your three vowels as a block. Ignore the ordering of the block for the moment. Now you have 6 remaining "letters"- the original consonants plus this "vowel block". Now, there are $6$ of these altogether. So you get $6!$ ways of ordering them- which is the standard permutation formula. Now for $each$ of these orderings, you can internally order the three vowels in the block. So you multiply by $3!$, which is the number of ways to order the vowels.

EDIT: The intuition for multiplication can be strengthened with some visualisation. Take a sheet of paper. Imagine all the "external" orderings of consonants and vowel block as being listed vertically. Now for each of these, write horizontally the 6 corresponding "internal" orderings of the vowels. You get a rectangular grid, each cell containing exactly one ordering. The area of this grid is clearly the breadth by its height. Hence $6!3!$.
