In how many different ways can the letters of the word ‘DAUGHTER’ be arranged so that the vowels always come together?

The given word contains 8 different letters.
When the vowels AUE are taken together, we may treat them as 1 letter.
Then,
The letters to be arranged are DGHTR (AUE)
The letters can be arranged in ⁶P₆ = 6!
= 720 ways.
The vowels AUE may be arranged in 3! = 6 ways.
Required number of ways = (720 × 6)
= 4320 ways.