What is...?

The Shannon entropy of a discrete random variable quantifies the number of bits of information conveyed by sampling that variable. Although originally introduced in the context of information theory, techniques relying on Shannon entropy have been used to prove results of very different flavors in fields such as combinatorics and number theory. I hope to build some intuition for Shannon entropy as a tool used to prove combinatorial results. I will present one or two of my favorite entropic proofs in combinatorics which showcase the power of this technique over more basic probabilistic or "averaging" arguments (which we will also see).