To the extent that "factorial" is an arbitrary function, you can define it any way that you like. This is trivially obvious, but not very useful.
But you will have a very hard time finding any mathematician who agrees that it either makes sense or is useful to define 0! as anything other than 1. I can think of a half-dozen reasons to make it 1. None to call it 0. If you did find such an odd mathematician, I guarantee that it won't be someone in combinatorics (which is the field of math where factorial tends to come up the most).
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