If there were not any duplicated (or triplicated) letters then there would be 6! or 720 ways to rearrange the letters. 6! = 6x5x4x3x2x1. 
This is because the first letter could be any of the 6 in the word tattoo. 
The 2nd letter could be any of the 5 remaining letters 
The 3rd could be any of the 4 remaining. 
The 4th could be any of the 3 remaining 
The 5th could be any of the 2 remaining 
The 6th has no choice = 1 

Therefore if all letters were different there would be 6x5x4x3x2x1 ways = 720