1. Count the letters:
* There are 12 letters in total.
2. Account for repetitions:
* "o" appears 4 times
* "a" appears 3 times
* "e" appears 2 times
3. Use the formula:
The number of arrangements (permutations) with repetitions is calculated as:
* n! / (n1! * n2! * ... * nk!)
Where:
* n is the total number of letters
* n1, n2, ... nk are the counts of each repeating letter
Applying the formula:
* n = 12
* n1 = 4 (for "o")
* n2 = 3 (for "a")
* n3 = 2 (for "e")
So the calculation is: 12! / (4! * 3! * 2!)
4. Calculate the result:
This gives us: 479,001,600
Therefore, there are 479,001,600 different ways to arrange the letters in the word "onomatopoeia".