Infinitely many. You could, for example, cut out images of the 8 letters and paste then on the wall facing you. Each letter can take infinitely many positions on that wall so you have infinitely many arrangements.
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There are 172 different arrangements.
There are 6! = 720 different arrangements.
In the word "function" you have 8 letters. 6 different letters and 2 equal letters.The number of different arrangements that are possible to get are:6!∙8C2 = 720∙(28) = 20 160 different arrangements.
There are 24.
There are 13 letters in "the world topic". This includes 2 ts and 2 os. Therefore there are 13!/[2!*2!] = 1556755200 different arrangements.