To solve these type of questions we must take all the vowels as one unit and do arrangements among all units and among the group which you have taken as on unit.

Approach:
To solve this problem, we can follow these steps:
1. Identify the vowels in the given word.
2. Find the number of ways the vowels can be arranged among themselves.
3. Treat the vowels as a single entity and arrange them with the consonants.
4. Find the number of ways the consonants can be arranged among themselves.
5. Multiply the results obtained in steps 2 and 4 to get the final answer.

Step-by-Step Solution:
1. The given word is 'CORPORATION'. The vowels in the word are O, O, A, I, O, and I. Let's denote them by V.
2. We have 6 vowels, which can be arranged among themselves in 6! ways.
3. Now, we treat the vowels as a single entity. So, we have 6 + 1 = 7 entities to arrange. Let's denote the consonants by C. So, we have VCVCVCVC.
4. We have 7 entities to arrange, out of which 3 are C's. So, we can arrange the consonants in 7!/(3!) ways.
5. Multiplying the results obtained in steps 2 and 4, we get the total number of arrangements as 6! x 7!/(3!) = 2,898,240.

Final Answer:
Therefore, the letters of the word 'CORPORATION' can be arranged in 2,898,240 different ways so that the vowels always come together.