In how many ways are the letters of the word DOCUMENT arranged so that all the vowels always come together?

The word DOCUMENT has the following letters:

Vowels: O, U, E (3 vowels)

Consonants: D, C, M, N, T (5 consonants)

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Since all vowels must come together, treat the vowels (O, U, E) as one single block.

This gives:

1 vowel block (OUE) + 5 consonants (D, C, M, N, T) = 6 units to arrange

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The 6 units can be arranged in:

$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1$

$= 720$ ways

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Within the vowel block, the 3 vowels can be arranged among themselves in:

$3! = 3 \times 2 \times 1$

$= 6$ ways

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Total arrangements = (Ways to arrange the 6 units) $\times$ (Ways to arrange vowels within the block)

Total $= 6! \times 3!$

$= 720 \times 6$

$= 4320$

Therefore, the letters of DOCUMENT can be arranged in 4320 ways with all vowels together.