MATH 204 • Midterm II • Discrete Mathematics

Basic Principles of Counting

Counting Examples

Permutations

Permutations Example

Groups and Circular Permutation Example

Identical Objects Example

Combination

n choose r

Committee Example

Ball Example

Binomial Theorem

Example 1

Example 2

Example 3

Pascal Identity

Vandermonde's Identity

Example 1

Combinatorial Proof

Example 1

Principle Inclusion-Exclusion

Example 1

Example 2

Distributing Objects into Boxes 1

Distributing Objects into Boxes 2

Example 1

Example 2

Example 3

Binomial Coefficients and Identities 1

Binomial Coefficients and Identities 2

Combinatorial Proof 1

Generalized Permutations and Combinations 1

Generalized Permutations and Combinations 2

Generalized Permutations and Combinations 3

Generalized Permutations and Combinations 4

Generalized Permutations and Combinations 5

Generalized Permutations and Combinations 6

Generalized Permutations and Combinations 7

Generalized Permutations and Combinations 8

Introduction

Homogenous Recurrence Relations

Example 1

Example 2

Example 3

Example 4

Non Homogenous Recurrence Relations

Example5

Example 6

Creating Recurrence Relations

Example 7

Example 8

Example 9

Recurrence Relations 1

Recurrence Relations 2

Recurrence Relations 3

Recurrence Relations 4

Recurrence Relations 5

Recurrence Relations 6

Recurrence Relations 7

Recurrence Relations 8

Recurrence Relations 9

Recurrence Relations 10

Recurrence Relations 11

Recurrence Relations 12

Recurrence Relations 13

Recurrence Relations 14

Generating Functions

Example 1

Example 2

Example 3

Example 4

Example 5

Extended Binomial Theorem

Counting Problems with Generating Functions

Example 6

Example 7

Example 8

Example 9

Example 10

Principle Inclusion - Exclusion

Example 1

Example 2

Generating Functions 1

Generating Functions 2

Generating Functions 3

Generating Functions 4

Generating Functions 5

Generating Functions 6

Generating Functions 7

Generating Functions 8

Generating Functions 9

Generating Functions 10

Generating Functions 11

Generating Functions 12

Generating Functions 13

Generating Functions 14

Generating Functions 15

Generating Functions 16

Generating Functions 17

Definition of Relation

Example 1

Example 2

Example 3

Example 4

Reflexive Relations

Example 5

Example 6

Symmetric Relations

Antisymmetric Relations

Example 7

Transitive Relations

Example 8

Properties of Relations

Example 9

Example 10

Example 11

Example 12

Equivalence Relation

Example 13

Example 14

Example 15

Composition of Relations

Example 16

Example 17

Properties of Relation 1

Properties of Relation 2

Equivalence Relation 1

Equivalence Relation 2

Equivalence Relation 3

Equivalence Relation 4

Equivalence Relation 5

Equivalence Relation 6

Equivalence Relation - Proof