Combinatorics: A Problem Oriented Approach( Classroom Resources Materials Series) / Edition 1

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Overview

This book teaches the art of enumeration, or counting, by leading the reader through a series of carefully chosen problems that are arranged strategically to introduce concepts in a logical order and in a provocative way. It is organized in eight sections, the first four of which cover the basic combinatorial entities of strings, combinations, distributions, and partitions. The last four cover the special counting methods of inclusion and exclusion, recurrence relations, generating functions, and the methods of Pólya and Redfield that can be characterized as "counting modulo symmetry." The unique format combines features of a traditional textbook with those of a problem book. The subject matter is presented through a series of approximately 250 problems, with connecting text where appropriate, and is supplemented by approximately 200 additional problems for homework assignments. Many applications to probability are included throughout the book. While intended primarily for use as the text for a college-level course taken by mathematics, computer science, and engineering students, the book is suitable as well for a general education course at a good liberal arts college, or for self study.
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Editorial Reviews

Ruth Michler
Combinatorics: A Problem Oriented Approach is a book on Combinatorics that mainly focuses on counting problems and generating functions. By restricting himself to an accomplishable goal, without attempting to be encyclopedic, the author has created a well-focused, digestible treatise on the subject. ...The choice of topics is excellent. This book is one of the few that treats the Pólya-Redfield counting method and recurrence relations.
MAA Online
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Product Details

  • ISBN-13: 9780883857106
  • Publisher: Mathematical Association of America
  • Publication date: 9/1/1998
  • Series: Classroom Resource Materials Series
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 156
  • Product dimensions: 5.90 (w) x 8.90 (h) x 0.40 (d)

Read an Excerpt

Strings
A string is a list, or sequence, of elements in a particular order. For the time being, we will concern ourselves with finite strings only.

Examples (1,3,0,1-12) is a string of integers. (X,Q,R,Z,X) is a string of letters from the usual alphabet {A,B, C,...,Z}

Notice that the same element can appear more than once in a string.
 When no confusion results, we will usually leave out the parentshes and commas in the representation of a string. The latter example above can be written more easily as XQRZX. This five-letter string can also be though of as a "word" that uses letters from the alphabet {A,...,Z}. The terms string, sequence, and word will be used interchangeably.

Example Any positive integer can be represented by a string of digits from the set {0,1,2,...,9}. This is the usual base 10 representation. In the binary system, any positive integer is represented by a string of digits (called bits in this case) from the set {0,1}.

Counting Strings
Continuing with the example above, consider all of the integers from 0 to 999. Each one of these can these can be represented by a 3-digit string using digits from 0 to 9; 0 is represented as 000, 1 as 001, etc. We know that there are 1000 of these integers (0 to 999), and the same result can be obtained can be obtained by counting 3-digit strings: The first digit can be selected in 10 different ways, and for each of 100, easy to select the first two digits. Finally, each of these 100 choice can be combined with one of the 10 choices for the third digit. The result is 1000 different three-digit strings.
 Here are some problems for you to try.

A1 By counting choices as in the examples above, determine the number of different five-bit strings, which are strings of 0s and 1s each consisting of five bits. What does this have to do with the binary representation of integers?

A2 Find the number of five letter words using letters of the alphabet {A,B,C,...,Z}
 These are instances of a more general problem, which can be summarized as follows.

Standard Problem #1
Find the number of strings of a given length that use elements from a given set.

The length of a string means the number of terms, or elements in the string. The answer to Standard Problem #1 is nk, where k  is the length of the string and n is the number of elements in the set from which the terms are selected.
 Next, we look at some variations on this standard problem. Suppose we want to count three-letter words that use letters from the usual alphabet, but with the condition that no letter can occur more than once in each word. Again, A2 there are 26 choice for the first letter. For each such choice there are only 25 letters that can be placed in the second position, because the letter used in the first position cannot be repeated. This 25o26 ways to place two different letters in the first two positions, and each of these choices can be combined with one of the remaining 24 letters in the third positions. The resulting number of three-letter words is 26o25o24.

A3 Find the number of five digit strings using digits from {0,1,21...,9} if no digit appears more than once in the string.

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Table of Contents

Part I Basics
Section A: Strings
Section B: Combinations
Section C: Distributions
Section D: Partitions

Part II Special Counting Methods
Section E: Inclusion and Exclusion
Section F: Recurrence Relations
Section G: Generating Functions
Section H: The Polya-Redfield Mathod

List of Standard Problems
Dependence of Problems
Answers to Selected Problems
Index

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