# Introduction

### From MM*Stat International

English |

Português |

Français |

Español |

Italiano |

Nederlands |

Combinatorial theory investigate possible patterns of orderings of finitely many elements, composed groups (sets) of such orderings, and the number of these orderings and groups.

## Different ways of grouping and ordering

Groups of elements can differ in several ways: they can contain either all elements just once, or some elements more times and others not at all; moreover, two groups that contain the same set of elements and differ from each other just by the ordering of the elements, can be considered to be the same or not.
**Examples with three elements a, b and c:**

- A group that contains every element exactly once:
*b c a* - A group that contains some elements more times and other elements not at all:
*b b* - Two groups that differ from each other just by the orderings of their elements:
*a b*and*b a*

As you can see on this simple example, groupings of elements can form three basic types:

## Use of combinatorial theory

Combinatorial theory mainly helps to answer questions such as:

- How many different ways can 5 different digits be ordered?
- How many ways exists for a choice of 10 words out of 30?
- How many possibilities are available for filling a lottery coupon?

Answers to these questions make possible to determine, for example, the probability of winning a lottery prize. Therefore, the use of combinatorial theory is most relevant in probability theory, which on the other hand, really use the results of combinatorial theory.