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One reason: for all numbers greater than zero, it’s true that N!/N = (N-1)!, and N!*(N+1) = (N+1)!. If 0! = 1, the pattern continues just fine, whereas picking any other number makes the behavior of the function strange.
A second reason: a factorial represents the total number of permutations possible for objects in a series. If you shuffle a deck of 52 cards, there are 52! possible resulting decks. If you shuffle a deck with 1 card, there’s only one possible order: the card by itself. If you have an empty box of cards, it likewise has a single “order”: nothing.
A third reason: even if you don’t accept the reasoning of shuffling an empty deck, there are other equations in combinatorics that only work if 0! = 1. It’s simpler to call it that by definition, than to rewrite all the combinatorics equations with special cases when one of the variables involved happens to be zero.
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