Find the number of ways to arrange the letters of the word OLYMPIAD such that the vowels (O, Y, I, A) always stay together. (Note: Treat Y as a vowel here as specified in this problem).

Step 1: The letters of the word OLYMPIAD are: O, L, Y, M, P, I, A, D. There are $8$ letters in total, all of which are unique.

• Vowels to stay together: $\{O,Y,I,A\}$. There are $4$ vowels.

• Consonants: $\{L,M,P,D\}$. There are $4$ consonants.

Step 2: Treat the group of vowels $\{O,Y,I,A\}$ as a single "super-letter" or block. We now have:

• The vowel block: $1$

• The $4$ individual consonants: $4$

Total elements to arrange: $1+4=5$ elements.

Step 3: The number of ways to arrange these $5$ elements is:

Step 4: Within the vowel block, the $4$ distinct vowels can be arranged among themselves in:

Step 5: The total number of arrangements is the product of these two counts:

So there are $2880$ valid arrangements.