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

To solve this problem, we need to find how many different ways the letters of the word "CORPORATION" can be arranged such that the vowels always come together.

### Step-by-step solution:

1. **Identify the vowels and consonants:**
   - The vowels in "CORPORATION" are: O, O, A, A, I.
   - The consonants are: C, R, P, R, T, N.

2. **Treat the vowels as a single unit:**
   Since the vowels must always come together, we can treat all 5 vowels (O, O, A, A, I) as one "block."

   So, we now have:
   - 6 consonants: C, R, P, R, T, N
   - Plus 1 "block" of vowels.

   This gives us a total of 7 units (6 consonants + 1 vowel block) to arrange.

3. **Arrange the consonants and vowel block:**
   These 7 units can be arranged in \( \frac{7!}{2!} \) ways. The division by \( 2! \) is because the letter 'R' repeats twice among the consonants.

   \[
   \frac{7!}{2!} = \frac{5040}{2} = 2520
   \]

4. **Arrange the vowels within the block:**
   Now, within the "block" of vowels, we have 5 vowels (O, O, A, A, I), with O and A repeating twice. The number of ways to arrange these vowels is:

   \[
   \frac{5!}{2!2!} = \frac{120}{4} = 30
   \]

5. **Final calculation:**
   The total number of ways to arrange the letters of the word "CORPORATION" such that the vowels always come together is the product of the two results:

   \[
   2520 \times 30 = 75,600
   \]

Thus, the number of different ways to arrange the letters of the word "CORPORATION" so that the vowels always come together is **75,600**.

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Would you like further details or have any questions?

Here are some related questions to deepen understanding:

1. How would the result change if the word had no repeated letters?
2. How do we adjust the calculation if only a subset of the vowels must stay together?
3. What if the vowels must be separated, rather than come together?
4. How would the calculation differ if the word had more repeated consonants?
5. Can you calculate the arrangements if consonants must also stay together?

**Tip:** When dealing with permutations of words with repeated letters, always divide by the factorial of the repetitions to avoid over-counting.

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

Explore how to arrange the letters of the word 'CORPORATION' so that the vowels always come together. This problem involves key concepts in combinatorics and permutations, making it suitable for students in Grades 9-12. The solution uses factorials and accounts for repeated letters to calculate the total number of ways.

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