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- Prove $0! = 1$ from first principles 18 answers
If I am correct in stating that a factorial of a number ( of entities ) is the number of ways in which those entities can be arranged, then my question is as simple as asking - how do you conceive the idea of arranging nothing ?
Its easy to conceive of a null element in the context of arrays, for example - so you say that there is only one way to present a null element.
But, in layman terms - if there are three humans $h1, h2, h3$ that need to be arranged to sit on three chairs $c1, c2, c3$ - then how do you conceive of a) a null human, and b) to arrange those ( that ? ) null humans ( human ? ) on the three chairs ?
Please note that referral to humans is just for easy explanation - not trying to be pseudo-philosophical. Three balls to be arranged on three corners of a triangle works just fine. So basically, how do you conceive of an object that doesn't exist, and then conceive of arranging that object ?
So, in essence ... is $0! = 1$, a convenience for mathematicians ? Not that its the only convenience, but just asking. Of course, there are many. If yes, then its a pity that I can't find it stated like so anywhere. If not, then can anybody suggest resources to read actual, good proofs ?