I was thinking about this after my professor mentioned the pigeonhole principle in class. Let's say we have $N$ items and $M$ containers. Here we assume $N > M$. We will randomly place each of the $N$ items into the $M$ containers. Is there a way to estimate the number of empty lists, as well as the length of the longest list?
I've tried looking up the pigeonhole principle online, but I could not find any algorithms or formulae.