.
How many different permutations of the letters in the word CINCINNATI are there?
~~~~~~~~~~~~~~~~~~~~~~~~~~
= 50400.
In all, there are 10 letters in the word.
Of them, C is repeated 2 times, I is repeated 3 times, and N is repeated 3 times.
The rest of the letters are unique.
The factors in the denominator, from left to right, are:
2! in the denominator is for C which is repeated 2 times in the word;
3! in the denominator is for I which is repeated 3 times in the word;
3! in the denominator is for N which is repeated 3 times in the word.