Waking up this morning, I was somehow reminded of this combinatorial identity that appeared on an exam in a “math problem solving” class I took, which I didn’t actually solve during the test because back then I was an idiot. It was
Basically, it’s observing that and then seeing that we have an instance of Vandermonde’s identity. The square is basically a form of obfuscation.
This stuff feels so obvious to me now yet it wasn’t back then. To make this entirely self-contained, I will prove Vandermonde’s identity as well for this specific case.