CONTENTS
1. Permutations and Combinations = 1
1.1. Two Counting Principles = 1
1.2. Ordered Choices = 4
1.3. Unordered Choices = 7
1.4. Circular and Restricted Arrangements = 11
1.5. Finite Probability = 14
1.6. Compound Events and Expectations = 18
1.7. Remarks = 23
2. The Binomial Theorem = 24
2.1. The Binomial Theorem = 24
2.2. Binomial Identities = 27
2.3. Multinomial Coefficients and Negative Binomial Coefficients = 33
2.4. Binomial Probabilities = 38
2.5. Remarks = 42
3. Enumeration = 44
3.1. Fibonacci Numbers = 44
3.2. Linear Diophantine Equations = 47
3.3. Inclusion - Exclusion = 50
3.4. Partitions = 55
3.5. Generating Functions = 57
3.6. Recurrence Relations = 62
3.7. Remarks = 69
4. Graphs = 70
4.1. Introduction = 70
4.2. Basic Concepts = 73
4.3. Eulerian Trails = 77
4.4. Hamiltonian Graphs = 81
4.5. Trees = 87
4.6. Binary Trees = 91
4.7. Remarks = 98
5. Finite Fields = 99
5.1. Introduction = 99
5.2. The Galois Field, G FP = 102
5.3. The Commutative Ring G FP ［x］ = 105
5.4. The Galois Field G F Pn = 108
5.5. Primitive Elements of G F Pn = 111
5.6. Operation in G F Pn = 113
5.7. Remarks = 118
6. Finite Plane Geometries = 119
6.1. The Geometry EG(2, Pn ) = 119
6.2. Some Properties of the Geometry EG(2, Pn ) = 124
6.3. The Projective Geometry PG(2, Pn ) = 130
6.4. Remarks = 134
7. Orthogonal Latin Squares and Error Correcting Codes = 135
7.1. Latin Squares = 135
7.2. Complete Sets of Orthogonal Latin Squares = 139
7.3. Error - Correcting Codes = 144
7.4. Hamming One - Error Correcting Binary Codes = 149
7.5. Remarks = 153
8. Balanced Incomplete Block Designs = 155
8.1. Relations Between Parameters = 155
8.2. Incidence Matrix of a BIB Design = 158
8.3. Symmetric BIB Designs = 163
8.4. Orthogonal Series Designs = 167
8.5. Symmetrically Repeated Differences = 170
8.6. Steiner Triples = 174
8.7. Symmetric BIB Designs with r ＝ (v - 1)／2 = 177
8.8. The Residual and Derived of a Symmetric BIB Design = 179
8.9. Remarks = 182
9. Problems of Choice = 184
9.1. Introduction = 184
9.2. The Pigeonhole Principle = 185
9.3. Ramsey's Theorem = 187
9.4. Erd o ·· s and Szekeres' Theorem = 192
9.5. Parity = 195
9.6. Remarks = 200
10. Optimization = 202
10.1. Introduction = 202
10.2. The Marriage Theorem = 205
10.3. Applications = 211
10.4. Algorithms for Matchings = 216
10.5. Maximum Flows = 220
10.6. Remarks = 231
Index = 233
