> Mathematically, `binomial(n, k)` for `k > n` is defined as 0.

It's not so clear cut. You can find different definitions out there. Knuth et. al., for example, in the book "Concrete Mathematics", extend the definition not just to negative k, but to negative n as well. Mathematicians aren't very good at agreeing on things. :-)

But that doesn't really matter: what we need to decide is what behaviour is useful for the users of the function.