# 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.

• @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. Jan 30, 2017 at 15:07
• What is a more mathematically intuitive explanation than "it follows by plugging in values into the recursive definition of the factorial"?
– MM8
Jan 30, 2017 at 15:09
• 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.
– user65203
Jan 30, 2017 at 15:12
• Empty products. Jan 30, 2017 at 15:22
• $6$ 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:$$()$$ Jan 30, 2017 at 15:43

• I do not think this is correct. The definition of $0!$ has nothing to do with arranging objects. Dec 20, 2022 at 23:29