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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.