Lattice chains and Delannoy paths represent two different ways to progress through a lattice. We use elementary combinatorial arguments to derive new expressions for the number of chains and the number of Delannoy paths in a lattice of arbitrary finite dimension. Specifically, fix nonnegative integers , and let denote the lattice of points that satisfy for . We prove that the number of chains in is given by where . We also show that the number of Delannoy paths in equals Setting (for all ) in these expressions yields a new proof of a recent result of Duchi and Sulanke relating the total number of chains to the central Delannoy numbers. We also give a novel derivation of the generating functions for these numbers in arbitrary dimension.