In how many different ways can the word "DAUGHTER" be arranged so that the vowels always come together?

The word "DAUGHTER" contains:

Vowels: A, U, E (3 vowels)

Consonants: D, G, H, T, R (5 consonants)

When vowels must always come together, treat them as a single unit.

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Consider the vowels (A, U, E) as one bundle. The arrangement now consists of:

(AUE), D, G, H, T, R

Total units to arrange $= 6$ units

Number of ways to arrange these 6 units:

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

$= 720$

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Within the vowel bundle, the 3 vowels can be arranged among themselves.

Number of ways to arrange A, U, E:

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

$= 6$

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Total arrangements $= 6! \times 3!$

$= 720 \times 6$

$= 4320$

Therefore, the word "DAUGHTER" can be arranged in $4320$ different ways with vowels together.