Illinois State University Mathematics Department

 MAT 312: Probability and Statistics for Middle School Teachers Dr. Roger Day (day@ilstu.edu)

### Counting Stategies: An Introduction to Combinatorics

Interest in computer science and the use of computer applications, together with connections to many real-world situations, have helped make topics of discrete mathematics more commonplace in school and college curricula. A topic of widespread application and interest is combinatorics, the study of counting techniques.
Enumeration, or counting, may strike one as an obvious process that a student learns when first studying arithmetic. But then, it seems, very little attention is paid to further developments in counting as the student turns to "more difficult" areas in mathematics, such as algebra, geometry, trigonometry, and calculus. . . . Enumeration [however] does not end with arithmetic. It also has applications in such areas as coding theory, probability, and statistics (in mathematics) and in the analysis of algorithms (in computer science). [Ralph P. Grimaldi, in Discrete and Combinatorial Mathematics, 1994, p. 3]

Combinatorial Analysis is an area of mathematics concerned with solving problems for which the number of possibilities is finite (though possibly quite large). These problems may be broken into three main categories: determining existence, counting, and optimization. Sometimes it is not clear whether a problem has a solution or not. This is a question of existence. In other cases solutions are known to exist, but we want to know how many there are. This is a counting problem. Or a solution may be desired that is "best" in some sense. This is an optimization problem. [John A. Dossey, Albert D. Otto, Lawrence E. Spence, & Charles Vanden Eynden, in Discrete Mathematics, 1987, p. 1]

Documents that support the reform of school mathematics education suggest the need for increased attention to topics in discrete mathematics as well as in probability and statistics. The topic of combinatorics--counting--is mentioned in the National Council of Teachers of Mathematics' Curriculum and Evaluation Standards for School Mathematics (1989) and in the Principles and Standards for School Mathematics: Discussion Draft (2000). The Mathematical Association of America's recommendations for teacher preparation identify combinatorics as a topic area worthy of study by middle school and high school teachers.

In this unit of the course and in the next one (on probability), we will study and apply combinatorial techniques in a variety of settings. As we study counting strategies and proability, look for connections to algebra, geometry, number theory, and many other topics in K-8 mathematics.

Here we will begin conceptualizing counting strategies that culminate in extensive use and application of permutations and combinations. We will refer several times to a menu at Blaise's Bistro, a mathematical restaurant devised for just those of us studying counting techniques.

The questions that follow all require that we count something, yet each involves a different approach.

#### The Pigeonhole Principle

We first explore a problem situation designed to highlight the Pigeonhole Principle, a fundamental counting strategy we make use of all the time, although you may never have called it by name prior to today.

How many students would be required to place soda orders, one soda per student, in order to insure that at least one of the six listed sodas would be ordered by at least two students?

The phrase "worst-case scenario" helps us determine the maximum number of students that could place orders without meeting the condition and then consider what must be done from there to meet the condition.

Next, consider two variables in the original problem situation: (1) the number of sodas available, and (2) the number of repeat orders desired. We can use n (here, n=6) to represent the first value and k (here, k=2) to represent the second. You are challenged to use n and k to write a statement that characterizes the principle used to solve the problem with these variables in place.

Does the following statement describe the principle behind the solution to the original problem, using the context of boxes and pigeons? Is so, how? If not, why not?

When placing pigeons into n boxes, the number of pigeons required to insure that at least one of the boxes has at least two pigeons is n+1 pigeons.

We can generate similar statements to describe the more general case with n boxes and at least k pigeons in one of the boxes. You may be asked to write and solve a problem, different from the sodas-and-students context, with which to test your general statements.

Here is one of the last pigeonhole problems we're likely to explore during today's session:

We have 10 boxes labeled 1 through 10 into which we place jelly beans. How many jelly beans are required to insure that at least one box contains at least as many jelly beans as the label on the box?

If I order one vegetable from Blaise's Bistro, how many vegetable choices does Blaise offer?

Here we select one item from a collection of items. Because there are no common items among the two sets Blaise has called Greens and Potatoes, we can pool the items into one large set. We use addition, here 4+5, to determine the total number of items to choose from.

If a choice from Group I can be made in n ways and a choice from Group II can be made in m ways, then the number of choices possible from Group I or Group II is n+m. Condition: None of the elements in Group I are the same as elements in Group II.

This can be generalized to a single selection from more than two groups, again with the condition that all groups, or sets, are disjoint, that is, have nothing in common.

Examples to illustrate the Addition Principle:

Here are three sets of letters, call them sets I, II, and III:

• Set I: {a,m,r}
• Set II: {b,d,i,l,u}
• Set III: {c,e,n,t}

How many ways are there to choose one letter from among the sets I, II, or III? Note that the three sets are disjoint, or mutually exclusive: there are no common elements among the three sets.

Here are two sets of positive integers:

• A={2,3,5,7,11,13}
• B={2,4,6,8,10,12}.

How many ways are there to choose one integer from among the sets A or B? Note that the two sets are not disjoint. What modification can we make to the Addition Principle to accommodate this case? Write that modification.

Multiplication Principle

A "meal" at the Bistro consists of one soup item, one meat item, one green vegetable, and one dessert item from the a-la-karte menu. If Blaise's friend Pierre always orders such a meal, how many different meals can be created?

We can enumerate the meals that are possible, preferably in some organized way to assure that we have considered all possibilities. Here is a sketch of one such enumeration, where {V,O}, {K,R}, {S,P,B,I}, and {L,A,C,F} represent the items to be chosen from the soup, meat, green vegetable, and dessert menus, respectively.

 VKSL VKPL VKBL VKIL ...and so on to... ORIL VKSA VKPA VKBA VKIA . ORIA VKSC VKPC VKBC VKIC . ORIC VKSF VKPF VKBF VKIF . ORIF
Note and be prepared to describe the enumeration process I have illustrated here.

How else could we complete the count without identifying all possible options? A map or tree to illustrate the enumeration process provides a bridge to such a method.

We have two ways to select a soup item, two ways to select a meat item, four green vegetables to choose from, and four desserts to choose from. The matching of one soup with each meat, then each of those pairs with each of four possible green vegetables, and each of those triples with each of four possible desserts leads to the use of multiplication as a quick way to count all the possible meals we could assemble at Blaise's.

This suggests we use the Multiplication Principle to describe this counting technique:

If a task involves two steps and the first step can be completed in n ways and the second step in m ways, then there are nm ways to complete the task. Condition: The ways each step can be completed are independent of each other.

This can be generalized to completing a task in more than two steps, as long as the condition holds.

Example to illustrate the Multiplication Principle:

Recall our three sets I, II, and III: {a,m,r}, {b,d,i,l,u}, and {c,e,n,t}. Determine the number of three-letter sets that can be created such that one letter is from set I, one letter in from set II, and one letter is from set III. Note that our choice in each set is independent of our choice in the other sets. If necessary, we could enumerate the possible three-letter, or three-element, sets.