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

In the word CORPORATION, we treat the vowels OOAIO as one letter.

Thus, we have CRPRTN (OOAIO). This gives us 7 (6 + 1) letters, of which R occurs 2 times and the rest are different.

Number of ways of arranging these letters = 7! / 2! = 2520.

Now, 5 vowels in which O occurs 3 times and the rest are different can be arranged in 5! / 3! = 20 ways.

∴ Required number of ways = 2520 × 20 = 50400.