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

The word CORPORATION has 11 letters with 5 vowels (O, O, A, I, O) and 6 consonants (C, R, P, R, T, N). Treating the 5 vowels as a single unit, we now have 7 units to arrange (6 consonants plus the vowel block). These 7 units contain the repeated letter R twice, so they can be arranged in 7 factorial divided by 2 factorial ways, equalling 2520. The 5 vowels within their block can be arranged in 5 factorial divided by 3 factorial ways (due to three O's), equalling 20. The total number of arrangements is 2520 times 20, which is 50400.

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