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

Keeping the vowels (AIA) together, we have CPTL (AIA).
We treat (AIA) as 1 letter.
Thus, we have to arrange 5 letters.
These can be arranged in 5! = (5 × 4 × 3 × 2 × 1) ways = 120 ways
Now, (AIA) are 3 letters with 2A and 1I
These can be arranged among themselves in
$$\frac{{3!}}{{2!}} = \frac{{3 \times 2 \times 1}}{{2 \times 1}} = 3$$     ways
∴ Required number of ways = 120 × 3 = 360