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

Problem restatement
Arrange 'CORPORATION' with all vowels in a single block. Account carefully for repeated letters.

Given data

• Letters (11 total): C(1), O(3), R(2), P(1), A(1), T(1), I(1), N(1).
• Vowels: O, O, O, A, I (5 vowels, with O repeated 3 times).
• Consonants: C, R, R, P, T, N (6 letters, with R repeated 2 times).

Concept/Approach
Treat the 5 vowels as a single block [V]. First, arrange the 7 items: [V] plus the 6 consonants (with R repeated). Then multiply by the internal arrangements of the vowels (with 3 O's repeated).

Step-by-step calculation
Arrange 7 items: [V], C, R, R, P, T, N ⇒ 7! / 2! (for the two R's) = 5040 / 2 = 2520 Arrange vowels inside [V]: 5! / 3! (for three O's) = 120 / 6 = 20 Total arrangements = 2520 × 20 = 50400

Verification/Alternative
Check counts: total letters 11; grouping vowels reduces to 7 items; duplicated R's and O's appropriately handled by dividing by factorials of repeats.

Common pitfalls

• Forgetting to divide by 2! for the two R's in the outer arrangement.
• Forgetting to divide by 3! for the three O's inside the vowel block.

Final Answer
50400