24
If we are looking at unique permutations, there would be half as many as there are stated above, as there are two o's in the word. So, in reality, there are only 12 permutations:
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10! permutations of the word "Arithmetic" may be made.
There are 7 factorial, or 5040 permutations of the letters in the word NUMBERS.
There are 195 3-letter permutations.
There are 5040.
239,500,800 12!/2! * * * * * Actually, as the word "permutation" [not permutations] has 11 letters, the answer is 11!/2! = 19,958,400