3! = 6
2! = 3! / 3 = (6) / 3 = 2
1! = 2! / 2 = (2) / 2 = 1
0! = 1! / 1 = (1) / 1 = 1
∴ 0! = 1
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0! is the number of bijections from an empty set to itself which is 1 😤
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Thats another great way to think about it. Is it more accurate or smth?
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it's just a direct definition that works in all cases and doesn't require an argument that it would be a good convention or would continue a pattern. in fact you can use this definition to prove that the pattern continues all the way down to zero
a much more mysterious "definition" that also accomplishes this is
n! = int_0^{\infty} x^n e^{-x} dx
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That is VERY mysterious to me 😄
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Word that mostly makes sense to me. I guess I wonder if it’s also just a convention to say factorials describe the number of set mappings? I’m only a lil baby discrete mathematician tho. Took one or two courses in university
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well in the sense that every definition is a convention. but that's one of the most important things factorials count (the other is the number of total orders and there's a bijection between these and permutations)
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