Count All Possible Unique Combinations of Letters in a Word
Python 2, 50 48 bytes



f=lambda s:s==''or len(s)*f(s[1:])/s.count(s[0])


Try it online!

No boring built-ins! To my surprise, this is even shorter than the brute force approach, calculating all of the permutations with itertools and taking the length.

This function uses the formula

\$\text{# of unique permutations} = \frac{\text{(# of elements)!}}{\prod_{\text{unique elements}}{\text{(# of occurences of that element)!}}}\$

and computes it on the fly. The factorial in the numerator is calculated by multiplying by len(s) in each function call. The denominator is a bit more subtle; in each call, we divide by the number of occurences of that element in what's left of the string, ensuring that for every character c, all numbers between 1 and the amount of occurences of c (inclusive) will be divided by exactly once. Since we divide only at the very end, we're guaranteed not to have any problems with Python 2's default floor division.