How many distinct arrangements of the letters in the word 'OPTICAL' can be made if all the vowels must be grouped together?

Explanation

The word 'OPTICAL' has 7 letters with vowels O, I, A. Treating the vowels as a single unit, we have 5 units to arrange (the vowel block plus the 4 consonants). The number of ways to arrange these units is 5! = 120. The vowels within the block can be arranged among themselves in 3! = 6 ways. Therefore, the total number of arrangements is 5! × 3! = 120 × 6 = 720.