# Basic Counting Techniques

Illinois State University Mathematics Department

MAT 305: Combinatorics Topics for K-8 Teachers

Basic Counting Techniques

The Addition Principle

The Multiplication Principle

Permutations

Combinations

Circular Permutations

Factorial Notation

Here we conceptualize some counting strategies that culminate in extensive use and application of permutations and combinations. The questions raised all require that we count something, yet each involves a different approach. The Addition Principle

If I order one vegetable from the menu at 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. This illustrates an important counting principle.

The Addition Principle

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. Necessary Condition: No 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? Try to write that modification. The 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

Take note of the enumeration process used in the table. How could you describe it in words? 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 another counting principle to describe this technique. The Multiplication Principle

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 aren*m ways to complete the task. Necessary 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...

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