To solve this problem, we will treat all the vowels in the word "COMPUTATION" as a single unit and then find the number of ways to arrange the remaining letters.

First, let's identify the vowels in the word: O, U, A, I, O. There are 5 vowels and 6 consonants (C, M, P, T, T, N).

Now, let's treat the group of vowels as a single unit: (OUAIO). We now have 7 units to arrange: C, M, P, T, T, N, and (OUAIO).

There are 7! (7 factorial) ways to arrange these 7 units:

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040

However, since there are two identical Ts, we need to divide by 2! to account for duplicate arrangements:

Adjusted arrangements = 5,040 ÷ 2! = 5,040 ÷ 2 = 2,520

Next, we need to consider the different ways the vowels can be arranged within their group. There are 5 vowels, so there are 5! ways to arrange them:

5! = 5 × 4 × 3 × 2 × 1 = 120

However, since there are two identical Os, we need to divide by 2! to account for duplicate arrangements:

Adjusted vowel arrangements = 120 ÷ 2! = 120 ÷ 2 = 60

Finally, we multiply the adjusted arrangements of the 7 units with the adjusted vowel arrangements:

Total arrangements = 2,520 * 60 = 151,200

So, there are 151,200 ways to arrange the letters of the word "COMPUTATION" such that all the vowels are together.
In how many ways can the letters of the word "COMPUTATION" be arranged such that all the vowels are together?