In how many different ways can the letters of the word "CORPORATION" be arranged so that all the vowels always come together?
The word "CORPORATION" has 11 letters with vowels O, O, A, I, O and consonants C, R, P, R, T, N.
1. Treat the vowels as a single unit, reducing the problem to 7 units (vowel group + 6 consonants).
2. Arrange these 7 units: \( \frac{7!}{2!} = 2520 \) (since R repeats).
3. Arrange the 5 vowels within their unit: \( \frac{5!}{3!} = 20 \) (since O repeats).
4. Multiply: \( 2520 \times 20 = 50,400 \).
Thus, the total arrangements keeping vowels together = 50,400.
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