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

Concept: When arranging n items in a row where some items are identical (repeated), the count of distinct arrangements is not n! -- that overcounts arrangements which only swap identical items among themselves. The permutations-with-repetition formula corrects for this: total distinct arrangements = n! / (r1! x r2! x ... x rk!), where r1, r2, ... are the repeat-counts of each item that appears more than once.

1. The word 'ECHRONICLEE' has 11 letters in total.

2. Among these, E appears 3 times and C appears 2 times; the remaining letters H, R, O, N, I, L each appear once.

3. Apply the formula: total arrangements = 11! / (3! x 2!).

4. 11! = 39,916,800; 3! = 6; 2! = 2, so the denominator is 6 x 2 = 12.

5. 39,916,800 divided by 12 = 3,326,400.

Cross-check (an independent method): choose positions for the three E's among the 11 slots -- C(11, 3) = 165 ways; from the remaining 8 slots, choose positions for the two C's -- C(8, 2) = 28 ways; arrange the remaining 6 distinct letters in the last 6 slots -- 6! = 720 ways. Multiplying: 165 x 28 x 720 = 3,326,400, confirming the result.

Total distinct arrangements = 3,326,400.