# Lecture Notes

WEEK # TOPICS
1

### Introduction (PDF)

• Platonic solids—Counting faces, edges, and vertices
• Planar graphs, duality
• Euler's formula for planar graphs—A constructive proof
• Non-existence of a sixth platonic solid
• Proving non-planarity by counting
2

### Counting 101 (PDF)

• 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

### Counting Sets (PDF)

• Set theory and Boolean logic
• Inclusion/exclusion—Easy as PIE
• How many handshakes?
4-5

### Counting Subsets (PDF)

• 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

### Discrete Probability (PDF)

• Sample spaces and events
• Probability measures
• Sampling with and without replacement
• Conditional Probabiliity
7

### More Probability (PDF)

• The Bernoulli process
• The Infinite Bernoulli process
• Analyzing games
• Solving problems
8

### Graph Theory (PDF)

• A whirlwind tour
• Vertices, edges, degree, paths, cycles
• Connectivity and components
• Acyclic graphs—Trees and forests
• Directed graphs
9

### More Graph Theory (PDF)

• Eulerian tours
• Graph coloring
• Ramsey Theory
• Turan's Theorem
10

### Contest Problems

• A chance to tackle some real contest problems