The sum I want to evaluate is $\sum_{i=2}^m \binom{p-1}{i-2} * \binom{n-k}{m-i}$. Here we have $p<k<=n$ and $m<=n$, all positive integers; I don't know how to communicate that to Wolfram Alpha though.
The correct answer (I think) is $\binom{n-k+p-1}{m-(k-p)+1}$.
Wolfram Alpha gives: $\frac{(-1)^m (-2 + k + m - n - p)!}{(-2 + m)! (k - n - p)!}$. Is this somehow equivalent to the correct answer? Why does it give factorials of negative integers? (admittedly, I didn't give Wolfram Alpha enough info to know these were negative integers, although $(k-n-p)!$ is pretty bad given I have $n-k$ as the top of a binomial coefficient in the sum).