The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is

We need to find how many words can be formed using all letters of "DAUGHTER" such that all vowels never come together.

DAUGHTER has 8 letters: D, A, U, G, H, T, E, R

Vowels: A, U, E (3 vowels)

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

All letters are distinct.

Total = $$8! = 40320$$

Treat the 3 vowels as a single unit. We then have 6 units (5 consonants + 1 vowel group) to arrange:

Arrangements = $$6! \times 3! = 720 \times 6 = 4320$$

(The $$3!$$ accounts for internal arrangements of the vowels within the group.)

Required count = Total - Vowels together = $$40320 - 4320 = 36000$$

The correct answer is Option 1: 36000.