What is the name of this expression?

Question

I'm trying to search this expression online, but don't know what it's called:

$$\frac{n!}{n^x(n-x)!} = \left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right) \cdots \left(1-\frac{x-1}{n}\right)$$

Thank you for your help.

Answer 1

If $x$ is an integer between $0$ and $n$, then that equality follows direct from the definitions. It has no special name. $$ (1-\frac{1}{n})(1-\frac{2}{n})\cdots(1-\frac{x-1}{n}) =\frac{(n-1)(n-2)\cdots(n-x+1)}{n^{x-1}} \\=\frac{n(n-1)(n-2)\cdots(n-x+1)}{n^{x}} =\frac{n(n-1)(n-2)\cdots(n-x+1)(n-x)!}{n^{x}(n-x)!} =\frac{n!}{n^x(n-x)!} $$