Chow rings of toric varieties feature a rich combinatorial structure of independent interest. We will begin by surveying four different ways of computing in these rings, due to Billera, Brion, Fulton–Sturmfels, and Allermann–Rau. We will illustrate the beauty and power of these methods by sketching four proofs of Huh and Huh–Katz’s formula μ_k(M) = deg_M(α*{r−k}β^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 α and β in the Chow ring of M. Each of these proofs sheds light on a different aspect of matroid combinatorics, and provides a framework for more recent developments in the intersection theory of matroids.