  Chow rings of toric varieties, which originate in intersection theory,
feature a rich combinatorial structure of independent interest. We survey four
different ways of computing in these rings, due to Billera, Brion,
Fulton--Sturmfels, and Allermann--Rau. We illustrate the beauty and power of
these methods by giving four proofs of Huh and Huh--Katz's formula $\mu^k(M) =
deg_M(\alpha^{r-k} \beta^k)$ for the coefficients of the reduced characteristic
polynomial of a matroid $M$ as the mixed intersection numbers of the hyperplane
and reciprocal hyperplane classes $\alpha$ and $\beta$ in the Chow ring of $M$.
Each of these proofs sheds light on a different aspect of matroid
combinatorics, and provides a framework for further developments in the
intersection theory of matroids. Our presentation is combinatorial, and does
not assume previous knowledge of toric varieties, Chow rings, or intersection
theory.
