How many different words can be made using the letters of the word…

Concept: When a specific group of letters in a word must always stay together (be adjacent) in every arrangement, treat that entire group as a single combined unit. Permute this unit together with the remaining letters, then separately multiply by the number of ways to arrange the letters inside the unit — using n!/(p1!·p2!·...) whenever letters repeat.

1. The word 'HALLUCINATION' has 13 letters: H, A, L, L, U, C, I, N, A, T, I, O, N.

2. Consonants: H, L, L, C, N, T, N — 7 letters, with L repeated twice and N repeated twice.

3. Vowels: A, U, I, A, I, O — 6 letters, with A repeated twice and I repeated twice.

4. Since all consonants must come together, treat the 7 consonants as a single block. This leaves 6 vowels + 1 block = 7 units to arrange.

5. Arrangements of these 7 units, correcting twice for the repeated A's and I's among the vowels: 7!/(2!×2!) = 5040/4 = 1260.

6. Arrangements of the 7 letters inside the consonant block, correcting twice for the repeated L's and N's: 7!/(2!×2!) = 5040/4 = 1260.

7. Total arrangements = 1260 × 1260 = 1,587,600.

Cross-check: The 7 arranging units (6 vowels + 1 consonant-block) plus the 7 letters unpacked inside that block account for all 13 original letters (6 + 7 = 13), and both repetition corrections (for the vowels' A/I and the consonants' L/N) are applied exactly once each — confirming no letter or repetition was double-counted or missed.