# About combinatorics

**Combinatorics** as a science began to speak in the 18th century. parallel with the emergence of the theory of probability, since for the solution of probabilistic problems it was necessary to calculate the number of different combinations of elements. With similar tasks, in which one or other subjects had to be selected, the best position of hunters during hunting, soldiers during the battle, people faced in the prehistoric era. Over time, there were various games (backgammon, cards, checkers, chess, etc.). In each of these games, we had to consider different combinations of figures, and the one who learned them better won, knew the winning combinations and knew how to avoid losing ones. Diplomats began to use ciphers based on combinatorial principles, for example, on various permutations of letters, and so on.

Some elements of combinatorics were known in India as far back as the 2nd century BC. Indians knew how to calculate numbers, which are now called "combinations". In the 12th century. Bhaskara calculated some kinds of combinations and permutations. It is believed that Indian scholars studied combinations in connection with their use in poetics, the science of the structure of verse, and poetic works. 2200 years ago the great Greek mathematician Archimedes wrote a treatise called "Stomachion". Scientists believe that this treatise was dedicated, no less than a combinatorial, i.e., science, which, as previously thought, the ancient Greek scientists did not yet know. The main goal of combinatorics is to study various combinations for solving a particular problem. And finding the number of ways in which the problem described in the "Stomachion" could be solved was so difficult that scientists had to check this decision with modern means. It took six weeks (!).

As a scientific discipline combinatorics was formed in the 17th century. The first scientific studies on combinatorics belong to the Italian scientist J. Cardano (1501-1576), N. Tartalier (1499-1557), G. Galilei (1564-1642) and the French scientist B. Pascal (1623-1662) and P. Fermat 1601-1665). At this time, the French mathematician Pierre Erigon obtained a formula for the number of combinations of n elements in m (without repetitions). The French author A. Takke devotes an entire chapter to the combinations and permutations in the book "Theory and practice of arithmetic" (1656).

Combinatorics Leibniz predicted a brilliant future and wide application. After the publication of Leibniz in 1665 of the work "Discourse on combinatorial art," the term "combinatorics" began to be used. In this book for the first time the scientific substantiation of the theory of combinations and permutations is given. But Leibniz's contribution is not limited to this: the scientist introduces special symbols and terms for this new area of mathematics, finds all the *k* -combinations of *n* -elements, outputs the combination properties, after which he discusses the application of combinatorics to logic, arithmetic, and even to problems of versification. Modern symbols of combinations (combination, placement, permutation: Ñ_{n}^{k}, À_{n}^{k}, Ð_{n}) was suggested by different authors of the training manuals only in the 19th century.

In modern society, with the development of computer technology about combinatorics, they spoke in a new way. The modern applied value of combinatorics can be seen in many areas. At present, the educational standard in mathematics includes the fundamentals of combinatorics, the solution of combinatorial problems by the method of enumeration, the compilation of a tree of variants. On combinatorics, the theory of games is based, and is used to create modern "intelligent" guidance and tracking systems. It is combinatorial methods that assesses the resistance to hacking of encryption and authentication systems both on the Internet and in military development. With combinatorics, the probability theory, as well as statistics and the social sciences based on it, have common problems. In modern combinatorics, and to this day, new discoveries are being made.