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This question already has an answer here:

I know that we define $n!$ as

$$n! = n\cdot(n-1)!,$$

so that $0! = 1$ follows from $1! = 1$.

However, what I would like to find out is the mathematical intuition behind $0! = 1$, if there is any.

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marked as duplicate by Dietrich Burde, Did, C. Falcon, Shobhit, E. Joseph Jan 30 '17 at 15:04

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ @EthanBolker Yes, I know. But seeing that a question is already a duplicate of a duplicate makes it perhaps more convincing that the question has been sufficiently discussed on MSE. $\endgroup$ – Dietrich Burde Jan 30 '17 at 15:07
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    $\begingroup$ What is a more mathematically intuitive explanation than "it follows by plugging in values into the recursive definition of the factorial"? $\endgroup$ – MM8 Jan 30 '17 at 15:09
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    $\begingroup$ Interestingly, the fact that the product of no factors is naturally defined to be $1$ appears to be less obvious than the sum of no term being defined to be $0$. But taking the logarithm explains it easily. $\endgroup$ – Yves Daoust Jan 30 '17 at 15:12
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    $\begingroup$ Empty products. $\endgroup$ – Simply Beautiful Art Jan 30 '17 at 15:22
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    $\begingroup$ $6$ ways to arrange three objects:$$\begin{matrix}(1,2,3)&(1,3,2)\\(2,1,3)&( 2,3,1)\\(3,1,2)&(3,2,1)\end{matrix}$$ $2$ ways to arrange two objects:$$\begin{matrix}(1,2)&(2,1)\end{matrix}$$ $1$ way to arrange one object:$$(1)$$ $1$ way to arrange zero objects:$$()$$ $\endgroup$ – Akiva Weinberger Jan 30 '17 at 15:43
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How many ways are there to arrange zero objects in a line? Only one way, the way that arranges no objects.

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    $\begingroup$ :D Well, that was a surprise answer $\endgroup$ – Simply Beautiful Art Jan 30 '17 at 15:22

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