In how many different ways can the letters of the word 'COORDINATION' be…

Concept: When specific letters of a word must always stay together, bundle those letters into a single block and treat that block as one unit alongside the remaining letters. The number of ways to arrange these n units is n!, divided by the factorial of the count of any letter that repeats among them; the letters inside the block can then be arranged independently among themselves, again dividing by the factorial of each letter that repeats within the block, and the two counts are multiplied together.

1. Split 'COORDINATION' into vowels and consonants. Vowels: O, O, O, I, I, A -- 6 letters in all, with O repeated 3 times and I repeated twice. Consonants: C, R, D, N, T, N -- 6 letters in all, with N repeated twice.

2. Since the vowels must always stay together, treat the group of 6 vowels as a single block. This leaves 6 consonants plus 1 vowel-block, i.e. 7 units, to be arranged.

3. Arrange the 7 units, dividing by 2! for the repeated N: 7!/2! = 2520.

4. Arrange the 6 vowels within their own block, dividing by 3! for the repeated O and by 2! for the repeated I: 6!/(3! x 2!) = 60.

5. Multiply the two independent counts, since each arrangement of the 7 units can be paired with each internal arrangement of the vowel block: 2520 x 60 = 151200.

Cross-check: The vowel block genuinely has only 60 distinct internal orders (not the full 6! = 720) because three of its letters are identical O's and two are identical I's, and the 7 units genuinely have only 2520 distinct external orders (not the full 7! = 5040) because two of them are identical N's; multiplying these two independently-verified counts gives 2520 x 60 = 151200.