# Why is $0! = 1$? [duplicate]

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.

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

• 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. – Yves Daoust Jan 30 '17 at 15:12
• $6$ 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:$$()$$ – Akiva Weinberger Jan 30 '17 at 15:43