TITLE: What are some other ways to compute the factorials using addition only?
QUESTION [8 upvotes]: As everyone here knows, or ought to know that multiplication is essentially just repeated addition.
Thus, for example, $$4! = 24$$ $$ = 6 + 6 + 6 + 6$$ $$ = (2 + 2 + 2) + (2 + 2 + 2) + (2 + 2 + 2) + (2 + 2 + 2).$$
The Tooth Fairy told me a fortnight ago that there is also a way to compute the factorial of a given number using additions only without knowing the answer beforehand (that is, unlike the method shown above).
I've tried adding up triangular numbers in various different ways and have come up empty. Or maybe it's with the squares (which in turn can be obtained by adding up odd numbers in order).
This riddle has me stumped.

REPLY [2 votes]: Take an $n$ letter word with distinct letters. Use casework to find the number of cases there are, adding by $1$ for each case. That number is $n!$.