What is the significance of $0!=1$?

Question

One can derive $0!=1$ by the formula

$\frac{n!}{(n-1)!}=n$

simply put $n=1$ and we get $0!=1$ .

But a question remains in my mind: what is its physical significance? For example:

$2!$ means: in how many ways we can arrange $2$ items?

$1!$ means we can arrange $1$ item in $1$ way.

Answer 1

I don't know that there is "physical" significance. If you use the combinatorial interpretation of counting permutations, then you run into the empty function as the only function $\varnothing\to\varnothing$, which is vacuously a permutation. Because of the vacuity, I don't really "see" it.

If we want a choice of $0!$ that makes all of our recursions, explicit formulas and summation notations with factorials to be organized uniformly and compactly then the only choice of $0!$ is $1$: that is the true significance of the choice $0!=1$.

• The defining recursion of course is $n!=n\cdot (n-1)!$, which we can use to "solve" for the appropriate choice of $0!$.
• The intuitive explicit formula $n!=n\cdot(n-1)\cdots 2\cdot 1$ when $n=0$ should arguably be interpreted as an empty product, a product of zero things. Given an associative binary operation $\circ$ and the associated $k$-ary operations $\square\circ\square\circ\cdots\circ\square$, the empty operation should be interpreted as the identity element of $\circ$ when there is one. This is "morally correct."
• Summations often involve factorials indexed from $n=0$. Taylor series and exponential generating functions are ubiquitous in math.

Related: my answer about $\binom{n}{k}$ when $(k,n)$ is not a pair of nonnegative integers with $k<n$.

Answer 2

First I'd say that deriving the value of $0!$ is nonsensical, as the meaning of factorials begins by defining $0! = 1$. Of course, it is chosen to be consistent with your formulation, as well as many others.

Now: Factorials can be interpreted as the number of ways to arrange $n$ objects in a sequence (permutations). If you have 0 objects then it's sensible to say that there's only one way to lay them out: the trivial empty sequence. Think cards: if you have zero cards then there is only one possible result of shuffling them and dealing them on a table -- the result will will definitely be an empty tabletop.

It's also consistent with the "empty product" principle, in that whenever you have no multiplications, the result should be 1. (Compare to exponents, etc.)

See also:

https://en.wikipedia.org/wiki/Factorial#Applications

https://en.wikipedia.org/wiki/Empty_product