Method of Trajectories in Combinatoric Problems

Abstract

The article describes the features of research work on
certain sections of probability theory. Geometric methods
that are often used in various mathematical sections of
algebra and the beginning of analysis are considered.
It is noted that the method is based on a simple idea of
geometric illustration of binomial coefficients.
This illustration makes it possible not only to illustrate
such a combinatorial structure as a combination, but also
to prove several important combinatorial identities. Many
problems of combinatorics are reduced to the calculation
of certain paths (trajectories) that have certain properties.
The often-mentioned ways are models of various practical
situations. The paper considers the application of the
trajectory method to prove combinatorial identities.
It is noted that to solve a combinatorial or
probabilistic problem, it is advisable to use its
geometric interpretation, which reduces the problem
to counting the number of paths (trajectories) with
certain properties. This is the method of trajectories.
The paper describes the application of the trajectory
method for solving combinatorial and probabilistic
problems.
The geometric interpretation of the main combinatorial
structures is given.
The theory of the trajectory method is analyzed,
examples of problem solving are given.
Combinatorial problems have been the subject of
many researchers in connection with the computerization
of all aspects of life. Computer models are discrete in
nature, which is why discrete mathematics has recently
developed rapidly. Combinatorics, as one of the branches
of discrete mathematics, attracts the attention of many
researchers due to its wide application in various fields of
knowledge and economics in particular.