> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/501-number-theory/630-combinatorics/counting-theory.md).

# Counting theory

* partitions
* permutations
* combinations

Counting is used to determine the number of objects with certain properties. For examples: The number of SIM cards a company has to produce if they've been assigned 7 digit numbers. Number of possible match starters (football, basketball) given the number of team members and their positions.

Basic counting rules

* Counting problem may be complex, so simplify it using decomposition.
* Basic decomposition rules:
  * *Product rule* 

    count decomposes into a sequence of dependent counts; each element in the first count is associated with all elements of the second count.
  * *Sum rule* 

    count decomposes into a set of independent counts; elements of counts are alternatives.

Techniques for determining, without direct enumeration, the number of possible outcomes of particular event or the number of elements in a set. Such counting is sometimes called *combinatorial analysis*. It includes the study of *permutations* and *combinations*.