Binomial Coefficients: Existence and Uniqueness by Pascal's Recursion, Vanishing above the Diagonal, Diagonal and First Values, the Factorial Formula and Symmetry

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let factorials be as in The Factorial of a Natural Number with Zero §factorial, and for $m\le n$ in $\mathbb{N}_{0}$ let $n-m$ be the difference.

¶ There is exactly one map $C:\mathbb{N}_{0}\times\mathbb{N}_{0}\to\mathbb{N}_{0}$ with $C(n,0)=1$, $C(0,k)=0$ for $k\ge1$, and $C(n+1,k+1)=C(n,k)+C(n,k+1)$, for all $n,k\in\mathbb{N}_{0}$.

Let $C$ be this map, and let $n,k\in\mathbb{N}_{0}$.

¶ If $k>n$, then $C(n,k)=0$.

¶ $C(n,n)=1$.

¶ $C(n,1)=n$.

¶ If $k\le n$, then $k!\,(n-k)!\,C(n,k)=n!$.

¶ If $k\le n$, then $C(n,k)=C(n,n-k)$.