Difference between revisions of "Combinatorics"
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| − | '''Combinatorics''' is the study of counting. | + | '''Combinatorics''' is the study of counting. Different kinds of counting problems can be approached by a variety of techniques. |
== Introductory combinatorics == | == Introductory combinatorics == | ||
| − | + | The following topics help shape an introduction to counting techniques: | |
| + | * [[Venn diagram]] | ||
* [[Combinations]] | * [[Combinations]] | ||
* [[Permutations]] | * [[Permutations]] | ||
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* [[Combinatorial identities]] | * [[Combinatorial identities]] | ||
* [[Binomial theorem]] | * [[Binomial theorem]] | ||
| + | |||
| + | === Lists -- the beginning === | ||
| + | Consider the task of counting the number of integers between 14 and 103 inclusive. We could simply list those [[integers]] and count them. However, we can renumber those integers so that they correspond to the [[counting numbers]] (positive integers), starting with 1. In this [[correspondence]], 14 corresponds to 1 (for the 1st integer in the list), 15 with 2, 16 with 3, etc. The relationship between the members of each pair is that the second is 13 less than the first. So, we we know that 103 corresponds to the 103 - 13 = 90th integer in the list. Thus the list is 90 integers long. | ||
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| + | Note that <math>13 = 14 - 1</math>, or 1 less than the first integer in the list. If we start our list with n and end with <math>m</math>, the number of integers in the list is | ||
| + | |||
| + | <math>\displaystyle m - (n -1) = m - n + 1.</math> | ||
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Revision as of 00:15, 19 June 2006
Combinatorics is the study of counting. Different kinds of counting problems can be approached by a variety of techniques.
Contents
Introductory combinatorics
The following topics help shape an introduction to counting techniques:
- Venn diagram
- Combinations
- Permutations
- Overcounting
- Complementary counting
- Casework
- Constructive counting
- Committee forming
- Pascal's triangle
- Combinatorial identities
- Binomial theorem
Lists -- the beginning
Consider the task of counting the number of integers between 14 and 103 inclusive. We could simply list those integers and count them. However, we can renumber those integers so that they correspond to the counting numbers (positive integers), starting with 1. In this correspondence, 14 corresponds to 1 (for the 1st integer in the list), 15 with 2, 16 with 3, etc. The relationship between the members of each pair is that the second is 13 less than the first. So, we we know that 103 corresponds to the 103 - 13 = 90th integer in the list. Thus the list is 90 integers long.
Note that 
, or 1 less than the first integer in the list. If we start our list with n and end with 
, the number of integers in the list is
Intermediate Combinatorics
- Principle of Inclusion-Exclusion
- Conditional Probability
- Recursion
- Correspondence
- Generating functions
- Partitions
