Combinatorial problem. The simplest combinatorial problems. Combinatorial problems: examples

Teachers of mathematics introduce their students to the notion of "combinatorial problem" in the fifth grade. This is necessary to ensure that they are able to continue to work with more complex tasks. Under the combinatorial nature of the problem, one can understand the possibility of solving it by searching the elements of a finite set.

The main feature of such an order is the question to them, which sounds like "How many options?" Or "How many ways?" The solution of combinatorial problems directly depends on whether they understood the decisive meaning, whether he was able to correctly represent the action or process that were described In the assignment.

How to solve a combinatorial problem?

It is important to correctly determine the type of all connections available in the problem, but it is necessary to check whether there are repeats of the elements in it, whether the elements themselves change, whether their order plays a big role, and also for some other factors.

Combinatorial task can have a number of restrictions that can be imposed on connections. In this case, you will need to calculate its solution completely and check whether these restrictions have any effect on the connection of all the elements. If the effect is really there, it is necessary to check which one.

Where to begin?

To begin with, you need to learn to solve the simplest combinatorial problems. Mastering with simple material will allow you to learn to understand more complex tasks. It is recommended that you first start solving problems with constraints that are not taken into account when considering a simpler version.

It is also recommended to try to solve first those problems in which fewer common elements need to be considered. Thus, you can understand the principle of creating samples and learn how to create them yourself. If the task for which it is necessary to use combinatorics consists of a combination of several simpler ones, it is recommended to solve it in parts.

Solution of combinatorial problems

Such tasks may seem simple in solution, but the combinatorics are rather difficult to master, some of them have not been solved for the last hundred years. One of the most famous tasks is to determine the number of magic squares of a special order, when the number n is greater than 4.

Combinatorial problem is closely related to the theory of probability, which appeared in medieval times. The probability of the origin of an event can only be calculated using combinatorics, in this case it will be necessary to alternate all the factors in places to obtain the optimal solution.

Problem Solving

Combinatorial tasks with the solution are used to train students and students to work with this material. If to speak in general, they should cause the person's interest and desire to find a common solution. In addition to mathematical calculations, it is necessary to apply mental stress and use a guess.

In the process of solving the tasks posed, the child will be able to develop mathematical imagination and combinatorial abilities, which can be of great use to him in the future. Gradually, the level of complexity of tasks to be solved must be increased in order not to forget the existing knowledge and add new ones to them.

Method 1. Bust

The methods for solving combinatorial problems are very different from each other, but they can all be used by the student to receive a response. One of the simplest, but at the same time, the longest methods is busting. With it, you just need to go through all the possible solutions without making any diagrams or tables.

As a rule, the question in this problem is connected with possible variants of the origin of an event, for example: what numbers can be made using the numbers 2, 4, 8, 9? Through a search of all options, a response is composed of possible combinations. This method is perfectly suitable if the number of possible options is relatively small.

Method 2. A tree of variants

Some combinatorial problems can be solved only by drawing up diagrams in which information about each element will be detailed. Drawing up a tree of possible variants is another way to find the answer. It is suitable for solving not too complicated tasks, in which there is an additional condition.

An example of such a problem:

• Which five-digit numbers can be made up of digits 0, 1, 7, 8? For the solution it will be necessary to build a tree from all possible combinations, and there is an additional condition - the number can not start from zero. Thus, the answer will consist of all numbers that will start with 1, 7 or 8.

Method 3: Creating tables

The solution of combinatorial problems can also be performed using tables. They are similar to the tree of possible options, because they offer a visual solution to the situation. To find the correct answer, you need to create a table, and it will be mirrored: the horizontal and vertical conditions will be the same.

Possible answers will be obtained at the intersection of columns and lines. In this case, the answers at the intersection of the column and the row with the same data will not be obtained, these crossings must be specially marked so as not to get confused when composing the final answer. This method is not too often chosen by students, many prefer a tree with options.

Method 4: Multiplication

There is another way with which you can solve combinatorial problems, the multiplication rule. It fits perfectly in the case when, by condition, you do not need to list all possible solutions, you just need to find their maximum number. This method is one of a kind, it is used very often when they start solving combinatorial problems.

An example of such a task might look like this:

• 6 people are waiting for the exam in the corridor. How many ways can I use to place them in a general list? To get an answer, it is necessary to specify how many of them can be in the first place, how many on the second, on the third, etc. The answer is the number 720.

Combinatorics and its types

Combinatorial task is not only a school material, university students also study it. In science, there are several types of combinatorics, and each of them has its own mission. The enumerative combinatorics should consider the tasks of enumerating and calculating possible configurations with additional conditions.

Structural combinatorics is a component of the university program, it studies the theory of matroids and graphs. Extreme combinatorics also has to do with university material, and here there are individual limitations. Another section is Ramsey's theory, which deals with the study of structures in random variations of elements. There is also a linguistic combinatorics, which deals with the question of the compatibility of certain elements among themselves.

The method of teaching combinatorial problems

According to the curriculum, the age of the pupils, which is designed for primary acquaintance with this material and for solving combinatorial problems, is class 5. It is there that for the first time this topic is offered to the students, they get acquainted with the phenomenon of combinatoriality and try to solve the tasks assigned to them. It is very important that when formulating the combinatorial problem a method is used, when the children themselves are engaged in the search for answers to questions.

Among other things, after studying this topic it will be much easier to introduce the concept of factorial and use it in solving equations, problems, etc. Thus, combinatoriality plays an important role in obtaining further education.

Combinatorial tasks: why are they needed?

If you know what combinatorial problems are, then you will not experience any difficulties with their decision. The method of their solution can be useful when it is necessary to create schedules, work schedules, and also complex mathematical calculations, for which the electronic devices do not work.

In schools with in-depth study of mathematics and computer science, combinatorial problems are studied additionally, special courses, methodological aids and tasks are compiled for this purpose. As a rule, several tasks of this type can be part of the Unified State Exam in Mathematics, usually they are "hidden" in part C.

How to solve a combinatorial problem quickly?

It is very important to be able to see the combinatorial problem quickly, because it can have a veiled formulation, this is especially important when passing the USE, where every minute counts. Write out separately the information that you see in the text of the task, on the sheet, and then try to analyze it in terms of four methods known to you.

If you can put information in a table or other education, try to solve it. If you can not classify it, in this case it is best to leave it for a while and move on to another task, so as not to waste precious time. This situation can be avoided if a number of tasks of this type are solved in advance.

Where to find examples?

The only thing that will help you learn to solve combinatorial problems is examples. You can find them in special mathematical collections that are sold in educational literature stores. However, there you can find information only for university students, schoolchildren will have to search for additional tasks, as a rule, for them tasks are invented by other teachers.

Teachers of universities believe that students need to train and constantly offer them additional educational literature. One of the best collections is "Methods of discrete analysis in solving combinatorial problems", written in 1977 and issued repeatedly by leading publishing houses of the country. It is there that you can find tasks that were relevant at that time and remain relevant today.

What if you need to compose a combinatorial problem?

Most often, combinatorial tasks must be made by teachers who are required to teach students to think unconventionally. Here everything will depend on the creative potential of the compiler. It is recommended to pay attention to already existing collections and try to compose a task so that it combines several methods of solving it at once and had different from the book data.

Teachers of universities in this respect are much freer than school ones, they often give their students the task to come up with combinatorial problems with detailed methods of solution and explanations. If you are not related to either, you can ask for help from those who really understand the matter, and also hire a private tutor. One academic hour is enough to make several similar tasks.

Combinatorics - the science of the future?

Many specialists in the field of mathematics and physics believe that it is the combinatorial problem that can be the impetus for the development of all technical sciences. It is enough to approach the solution of various problems unconventionally, and then it will be possible to answer questions that have been haunting scientists for several centuries. Some of them seriously assert that combinatorics is a help for all modern sciences, especially cosmonautics. It will be much easier to calculate the flight trajectories of ships with the help of combinatorial problems, and they will also help to determine the exact location of certain heavenly bodies.

The implementation of the non-standard approach has long started in the Asian countries, where the students solve even the elementary problems of multiplication, subtraction, addition and division, using combinatorial methods. Surprisingly many European scientists, the technique really works. Schools in Europe have just begun to learn from their colleagues. When combinatorics becomes one of the main sections of mathematics, it is difficult to foresee. Now science is being studied by the world's leading scientists who seek to popularize it.