**Combinatorics** is an area of discrete mathematics that studies collections of distinct objects and the ways that they can be counted or ordered, or used to satisfy some optimality criterion.

The most basic ideas in combinatorics include:

- factorials
- The number of possible arrangements of $ n $ distinct items is
, written $ n! $ , which equals*n*-factorial

- $ n\times(n-1)\times(n-2)\times\cdots\times2\times1 $

- Example: Three items, A, B, and C, can be arranged in $ 3!=3\times2\times1=6 $ different ways: ABC, ACB, BAC, BCA, CAB, and CBA.

- permutations
- The number of arrangements that are possible when a subset of $ k $ items is taken from a set of $ n $ distinct items is a "
**permutation of $ n $ objects taken $ k $ at a time**", which can be written as $ P^n_k $ or $ {}_nP_k $ , and is equal to $ \frac{n!}{(n-k)!} $ .- Example: The number of possible arrangements of the four letters A, B, C, D, taken two at a time, is $ P^4_3=\frac{4!}{(4-2)!}=\frac{24}{2}=12 $: AB, BA, AC, CA, AD, DA, BC, CB, BD, DB, CD, and DC.

- combinations
- The number of possible subsets of $ k $ items taken from a set of $ n $ items, where the order of the items doesn't matter (e.g., the sets ABC and BCA are considered equivalent), is a "
**combination of $ n $ objects taken $ k $ at a time**", which is written $ C^n_k $ or $ {}_nC_k $ or $ \binom{n}{k} $ , and is equal to $ \frac{n!}{k!(n-k)!} $ .- Example: The number of subsets of two letters chosen from the four letters A, B, C, and D, is $ C^4_2=\frac{4!}{2!(4-2)!}=\frac{24}{2\times2}=6 $ : AB, AC, AD, BC, BD, and CD.

- distributions
- partitions of integers or of sets
- recurrence relations
- inclusions
- inversions
- inclusion/exclusion principle
- derangements and subfactorials
- repetitions and replacements
- various restrictions placed on problems
- fundamental counting principle
- circular permutations
- generating functions
- free and fixed permutations, rotational symmetry and reflective symmetry
- cyclic permutations
- multisets
- Pascal's Triangle