1   Platonic solids—Counting faces, edges, and vertices
 Planar graphs, duality
 Euler's formula for planar graphs—A constructive proof
 Nonexistence of a sixth platonic solid
 Proving nonplanarity by counting

2   First Law of Counting—Multiplying the possibilities
 Shepard's Law—To count the sheep, count the feet
 Counting by cases—Break it down and add it up
 Counting by subtraction—Cases to exclude

3   Set theory and Boolean logic
 Inclusion/exclusion—Easy as PIE
 How many handshakes?

45   Binomial coefficients
 The wonders of Pascal's triangle
 Counting by block walking
 Counting by committee
 The most useful combinatorial identity known to man—"The Hockey Stick"
 The n days of Christmas
Problem Solving  Applying what we know—Examples of counting
 How to recognize an apparently unfamiliar problem
 What to do when you are lost in the forest—How to get unstuck
 More examples—kids and candy, flower arrangements

6   Sample spaces and events
 Probability measures
 Sampling with and without replacement
 Conditional Probabiliity

7   The Bernoulli process
 The Infinite Bernoulli process
 Analyzing games
 Solving problems

8   A whirlwind tour
 Vertices, edges, degree, paths, cycles
 Connectivity and components
 Acyclic graphs—Trees and forests
 Directed graphs

9   Eulerian tours
 Graph coloring
 Ramsey Theory
 Turan's Theorem

10  Contest Problems  A chance to tackle some real contest problems
