In how many different ways can the letters of the AUCTION be arranged in such a way that the vowels always come together?

The given word contains 7 different letters.
Keeping the vowels (AUIO) together, we take them as 1 letter.
Then,
we have to arrange the letters CTN(AUIO).
Now, 4 letters can be arranged in 4! = 24 ways.
The vowels (AUIO) can be arranged themselves in 4! = 24 ways.
∴ Required number of ways = (24 × 24)
= 576.