# 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*. --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/501-number-theory/630-combinatorics/counting-theory.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.