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# Mathematics of Choice: How to count without counting

Ivan Niven
Volume: 15
Edition: 1
Pages: 215
Stable URL: http://www.jstor.org/stable/10.4169/j.ctt19b9k1k

1. Front Matter (pp. i-vi)
3. Preface (pp. xi-xii)
4. CHAPTER ONE Introductory Questions (pp. 1-6)

The purpose of this chapter is to present a few sample problems to illustrate the theme of the whole volume. A systematic development of the subject is started in the next chapter. While some of the sample questions introduced here can be solved with no theoretical background, the solution of others must be postponed until the necessary theory is developed.

The idea of this book is to examine certain aspects of the question “how many?”. Such questions may be very simple; for example, “How many pages are there from page 14 to page 59, inclusive?” In some cases, the answer...

5. CHAPTER TWO Permutations and Combinations (pp. 7-26)

This chapter and the next introduce some of the fundamental ideas of the subject of this book. The reader may recognize a number of these concepts from previous study. However, at several places in Chapters 2 and 3 the topics are discussed in more detail than is usually the case in elementary books on algebra. It will smooth the way for the reader in subsequent chapters if he fully understands these fundamental ideas. If he is able to answer the questions in the problem sets, he can be sure of his understanding of the subject. Much of the basic notation...

6. CHAPTER THREE Combinations and Binomial Coefficients (pp. 27-49)

There are other ways, besides those in the preceding chapter, of looking atC(n, r), the number of combinations ofndifferent things takenrat a time. Several of these possibilities are studied in this chapter. We begin by pointing out that we can easily solve the path problem which was listed as Problem 1.3 in Chapter 1. For convenience, we repeat the statement of the question.

A man works in a building located seven blocks east and eight blocks north of his home. Thus in walking to work each day he goes fifteen blocks. All the streets in...

7. CHAPTER FOUR Some Special Distributions (pp. 50-66)

Many a problem in combinatorial analysis is solved by first reformulating it. This point was illustrated on page 28 where a path problem was reduced to an equivalent question involving combinations. In this chapter we look at some other problems which, when viewed from the proper perspective, also reduce to questions involving combinations.

Consider the question: In how many ways can eight plus signs and five minus signs be lined up in a row so that no two minus signs are adjacent? An example of such an arrangement is:

+ + − + − + + + − + −...

8. CHAPTER FIVE The Inclusion-Exclusion Principle; Probability (pp. 67-90)

In this chapter we prove a theorem of a very broad kind and then apply it to particular problems. The idea ofprobabilityis introduced towards the end of the chapter.

It will be convenient to lead up to the inclusion-exclusion principle by a sequence of three problems listed in increasing order of difficulty. The first problem is not very difficult at all.

Problem 5.1 How many integers between 1 and 6300 inclusive are not divisible by 5? Since precisely every fifth number is divisible by 5, we see that of the 6300 numbers under consideration, exactly 6300/5 or 1260...

9. CHAPTER SIX Partitions of an Integer (pp. 91-99)

In this chapter we discuss partitions of an integer, or what is the same thing, partitions of a collection of identical objects. In case the objects are not identical, the problem comes under the heading “partitions of a set” and is discussed in Section 8.2.

The partitions of a positive integer are the ways of writing that integer as a sum of positive integers. The partitions of 5, for example, are

5$\begin{matrix} 4 \quad+ \quad1 \\ 3 \quad+ \quad2 \end{matrix}\; \; \; \; \; \; \; \begin{matrix} 3 \quad+ \quad1 \quad+ \quad1 \\ 2 \quad+ \quad2 \quad+ \quad1 \end{matrix}\; \; \; \; \; \; \; \begin{matrix} 2 \quad+ \quad1 \quad+ \quad1 \quad+ \quad1 \quad \\ 1 \quad+ \quad1 \quad+ \quad1 \quad+ \quad1 \quad+ \quad1 \end{matrix}$

Since there are seven partitions of 5, we writep(5) = 7; in general, we letP(n) denote the number of partitions of the positive integer...

10. CHAPTER SEVEN Generating Polynomials (pp. 100-108)

In this chapter we shall use polynomials to “generate” the solutions of a class of problems. For example, we shall solve Problem 1.5 of Chapter 1: In how many ways is it possible to make change for a dollar bill? The method introduced in this chapter is, in its level of sophistication, just one step above the enumeration of cases.

In order to find the number of ways of changing a dollar bill, we first examine the well-known technique of multiplying polynomials. We shall be concerned, in particular, with multiplying polynomials whose coefficients are 1. For example,

$(1+x+x^{2}+x^{4}+x^{8})(1+x^{3}+x^{6}+x^{9})$

$=1+x+x^{2}+x^{3}+2x^{4}+x^{5}+x^{6}+2x^{7}$...

11. CHAPTER EIGHT Distribution of Objects Not All Alike (pp. 109-119)

Many problems of combinatorial analysis can be stated in terms of the number of ways of distributingobjectsinboxes. Some of these distribution problems were considered in earlier chapters. We now make a brief classification of the various types of questions.

First, the objects may be considered to be alike, and the boxes also indistinguishable from one another. These are partition problems. For example, the number of ways of distributing nine objects in four boxes is the same as the number of partitions of 9 into at most four summands. Problems of this sort were discussed in Chapters 6...

12. CHAPTER NINE Configuration Problems (pp. 120-128)

The questions discussed in this chapter are related to geometric patterns or configurations of one kind or another. We begin with a concept which is widely used throughout mathematics—the pigeonhole principle.

If eight pigeons fly into seven pigeonholes, at least one of the pigeon holes will contain two or more pigeons. More generally, ifn+ 1 pigeons are innpigeon holes, at least one of the holes contains two or more pigeons.

This simple form of the pigeonhole principle can be generalized as follows: If 2n+ 1 pigeons are innpigeonholes at least one of...

13. CHAPTER TEN Mathematical Induction (pp. 129-139)

Consider the sums of the odd integers:

(10.1) 1 = 1

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

1 + 3 + 5 + 7 + 9 = 25

1 + 3 + 5 + 7 + 9 + 11 = 36

A clear pattern emerges in the sums 1, 4, 9, 16, 25, 36; they are the squares of the natural numbers 1,2,3,4,5,6. These equations suggest the general proposition that the sum of the firstnodd positive integers is equal ton2, or, stated...

14. CHAPTER ELEVEN Interpretations of a Non-Associative Product (pp. 140-152)

Consider the mathematical expression

$2^{3^{4}}$.

There appear to be two ways of interpreting this—one by starting with 23and so interpreting the expression as 84; another by starting with 34and so interpreting the expression as 281. These ways lead to different results because 84is 4096 whereas 281is much larger. We can indicate these two interpretations by using parentheses; thus

(11.1)$(2^{3})^{4}=8^{4},\; \; \; 2^{(3^{4})}=2^{81},\; \; \; \mathrm{and}\; \; \; (2^{3})^{4}\neq 2^{(3^{4})}$.

Now in actual fact there is a convention or agreement in mathematics as to precisely how$2^{3^{4}}$is to be interpreted, namely as the second form in (11.1),

$2^{3^{4}}=2^{(3^{4})}=2^{81}$.

For the purposes...

15. Miscellaneous Problems (pp. 153-159)
16. Answers and Solutions (pp. 160-198)
17. Bibliography (pp. 199-200)
18. Index (pp. 201-202)