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

4320

The correct answer is Option (2) → 4320

Word: DAUGHTER

Total letters = 8

Vowels in "DAUGHTER" = A, U, E → 3 vowels

Consonants = D, G, H, T, R → 5 consonants

Vowels must come together → Treat vowels (AUE) as a single unit

Then total units = 5 consonants + 1 vowel block = 6 units

These 6 units can be arranged in $6!$ ways

Within the vowel block, vowels A, U, E can be arranged among themselves in $3!$ ways

Total arrangements = $6! \times 3! = 720 \times 6 = 4320$