Peter Shor teaches the fourth lecture of MIT's 18.200 Principles of Discrete Applied Mathematics, covering combinatorial counting techniques. He works through binomial coefficients and bijective proofs, demonstrating the method of counting the same set two different ways to establish identities. The bulk of the lecture builds toward Catalan numbers, showing how the same sequence counts Dyck paths, plane trees, and binary trees, three objects that look unrelated until Shor constructs explicit bijections between them. The session runs seventy eight minutes in a standard blackboard lecture format, with Shor developing each argument step by step rather than presenting finished results. This is core material for discrete mathematics, useful for anyone studying combinatorics, algorithm analysis, or the structural counting arguments that show up throughout computer science and discrete applied math.