Preface 1 Let s Count! 1.1 A Party 1.2 Sets and the Like 1.3 The Nunmber of Subsets 1.4 The Approximate Number of Subsets 1.5 Sequences 1.6 Permutations 1.7 The Number of Ordered Subsets 1.8 The Number of Subsets of a Given Size 2 Combinatorial Tools 2.1 Induction 2.2 Comparing and Estimationg numbers 2.3 Inclusion-Exclusion 2.4 Pigeonholes 2.5 The Twin Paradox and the Good Old Logarithm 3 Binomial Coefficients and Pascal s Triangle 3.1 The Binomial Theorem 3.2 Distributing Presents 3.3 Anagrams 3.4 Distributing Money 3.5 Pascal s Trianglc 3.6 Identities in pascal s Triangle 3.7 A Bird s -Eye View of Pascal s Triangle 3.8 All Eagle s -Eye View:Fine Details 4 Fibonacci Numbers 4.1 Fibonacci s Exercise 4.2 Lots of Identities 4.3 A Formula for the Fibonacci Nunbers 5 Combinatorial Probability 5.1 Events and Probabilities 5.2 Independent Repetition of an Experiment 5.3 The Law of Large Numbers 5.4 The Law of Small Numbers and t he Law of Very Large Nmmbers 6 Integers,Divisors and Primes 6.1 Divisibility of Integers 6.2 Primes and Their History 6.3 Factorization into Primes 6.4 On the Set of primes 6.5 Fermat s Little Theorem 6.6 The Fuclidean lgorithm 6.7 Congruences 6.8 Strange Numbers 6.9 Nunber Theory and Combiatorics 6.10 How to Test Whether a Number is a Prime? 7 Graphs 7.1 Even and Odd Dergrees 7.2 Paths Cycles and Connectivitry 7.3 Eulerian Walkd and Hamiltnian Cycles 8 Trees 8.1 How to Define Trees 8.2 How to Grow Trees 8.3 HOw to Count Trees? 8.4 How to Store Trees 8.5 The Number of Unlabeled Trees 9 Finding the Optimum 9.1 Finding the Best Tree 9.2 The Traveling Salesman Problem 10 Matvchings in Graphs 10.1 A Dancing Problem …… 11 Combinatorics in Geometry 12 Euler s Formula 13 Coloring Maps and Graphs 14 Finite Geometries,Codes,Latin Squares,and Other Pretty Creatures 15 A Glimpse of COmplexity and Cryptography 16 Answers to Exercises Index
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