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

Sure, here is the step by step solution:

1. First, let's identify the vowels and consonants in the word 'CORPORATION'. The vowels are O, O, A, I, O and the consonants are C, R, P, R, T, N.

2. When the question says that vowels should always come together, we can suppose all the vowels as a single letter. So, the word becomes CRPRTN (OOAIO). Here, we have 6 letters (including the group of vowels as one letter).

3. These 6 letters can be arranged in 6! ways. However, we have the letter 'R' repeating twice. So, we divide by 2! to account for this repetition. This gives us 6!/2! arrangements.

4. Now, within the group of vowels (OOAIO), we have 5 letters which can be arranged in 5! ways. But, we have the letter 'O' repeating three times. So, we divide by 3! to account for this repetition. This gives us 5!/3! arrangements.

5. Since these are independent events, we multiply the two results together to get the total number of arrangements. So, the total number of ways the letters of the word 'CORPORATION' can be arranged so that the vowels always come together is (6!/2!) * (5!/3!) = 720 * 20 = 14400 ways.