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
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