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

Problem restatementTreat the vowels in 'OPTICAL' as a single block so that they always stay adjacent, and count the total distinct arrangements.

Given data

• Word: OPTICAL (7 distinct letters).
• Vowels: O, I, A (3 vowels).
• Consonants: P, T, C, L (4 consonants).

Concept/ApproachGroup the 3 vowels as one block [V]. Then arrange [V] with the 4 consonants (total 5 items), and finally permute the vowels inside [V].

Step-by-step calculation Arrange 5 items ([V], P, T, C, L): 5! = 120 Permute vowels within [V] (O, I, A): 3! = 6 Total arrangements = 120 × 6 = 720

Verification/AlternativeNo repeated letters, so no division by factorials for duplicates. The block method is exact.

Common pitfalls

• Forgetting to multiply by the internal permutations of the vowels.
• Accidentally treating any letters as repeated (they are all distinct here).

Final Answer720