$n$ choose $k$ where $n$ is less than $k$

I am working on parameter estimation and one of the estimators involves a summation of $_nC_k$ ($n$ choose $k$) expressions. For some iterations, I need to compute expressions like $_0C_1$, $_0C_2$, etc.

In general how do we compute $_nC_k$ when $n$ is less than $k$? Do we still use the formula $\frac{n!}{(n-k)!k!}$ and use the gamma function to compute the negative factorial?

Thanks!

You could define $$_nC_k \text{ or } {n \choose k} = \frac{n(n-1)\cdots(n-k+2)(n-k+1)}{k(k-1)\cdots 2\cdot 1}$$ for positive integer $k$, and ${n \choose 0} = 1$ as the quotient of empty products.

This would give the usual values for non-negative integer $n \ge k$.

It would also give values for other real $n$. For non-negative integer $n \lt k$ including ${0 \choose k}$ it would give $0$ as the numerator involves multiplication by $0$ while the denominator does not.