N Choose Negative Number is Zero

Theorem
Let $n \in \Z$ be an integer.

Let $k \in \Z_{<0}$ be a (strictly) negative integer.

Then:
 * $\dbinom n k = 0$

Proof
From Pascal's Rule we have:
 * $\forall n, k \in \Z: \dbinom n {k - 1} = \dbinom {n + 1} k - \dbinom n k$

Thus it is sufficient to prove that:
 * $\forall n \in \Z: \dbinom n {-1} = 0$

So:

Hence the result.