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0!= 1 ?!


Tnad

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Intuitively: how many ways do think there are to order zero objects?

 

Slightly less intuitively:

[math]n! = n (n-1)![/math]

 

[math]1! = 1 \cdot 0![/math]

 

[math]0! = \tfrac{1!}{1} = 1[/math]

 

Really: that's just the definition.

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Oh I just thought, it's worth mentioning that the factorial function can be generalised to the Gamma function.

 

[imath]n! = \Gamma(n+1)[/imath] where [imath]\Gamma(z) := \int_0^\infty t^{z-1} e^{-t}\,dt[/imath].

 

And you should be able to verify that [imath]\Gamma(1)=\Gamma(2)=1[/imath] if you're okay with integration by parts.

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