Here are the steps in determining the distinguishable permutation of a given word.
1. Count the total numbers in a word. For the word TALLAHASSEE, there are a total of 11 letters.
2. List down the unique letters and their frequencies or how many times they occurred.
T - 1 time only
A - 3 times
L - 2 times
H - 1 time
S - 2 times
E - 2 times
3. Get the product of the factorial of each frequency per letter.
[tex]1!\times3!\times2!\times1!\times2!\times2![/tex][tex]1\times6\times2\times1\times2\times2=48[/tex]4. Divide the factorial of the total number of letters by the result in step 3.
[tex]11!\div48=831,600[/tex]Hence, there are a total of 831, 600 distinguishable permutations for the word TALLAHASSEE.
For the word MISSISSAUGA, there are a total of 11 letters too. These are:
1 Letter M
2 Letter I
4 Letter S
2 Letter A
1 Letter U
1 Letter G
So, similar to the previous word, we multiply the product of the factorial of each frequency per letter.
[tex]1!\times2!\times4!\times2!\times1!\times1![/tex][tex]1\times2\times24\times2\times1=96[/tex]Then, divide the factorial of the total number of letters which is 11! by 96.
[tex]11!\div96[/tex][tex]39,916,800\div96=415,800[/tex]Hence, there are 415,800 distinguishable permutations of the word MISSISSAUGA.
