How many arrangements are possible using the letters in the word FUZZY if each letter “Z” is distinctly different than the other? How many arrangements are possible if the letter “Z” is interchangeable with the other? Explain your reasoning.
If the letter "Z" is distinct from the other letters, then there are a total of 5 letters in the word "FUZZY", with 2 repeating letters "Z". Therefore, the total number of arrangements is
5! / (2!2!) = 60 arrangements
If the letter "Z" is interchangeable with the other letters, then there are only 4 unique letters in the word "FUZZY", with 2 repeating letters "Z". Therefore, the total number of arrangements is
4! / 2! = 12 arrangements
The reasoning behind the differing number of arrangements is due to the fact that when "Z" is interchangeable with the other letters, there are fewer unique elements to permute, resulting in fewer overall arrangements.