Mathematics teachers introduce their students to the concept of "combinatorial problem" as early as the fifth grade. This is necessary in order for them to be able to work with more complex tasks in the future. The combinatorial nature of a problem can be understood as the possibility of solving it by enumeration of elements of a finite set.
The main sign of tasks of this order is the question to them, which sounds like “How many options?” or "In how many ways?" The solution of combinatorial problems directly depends on whether the person who solved them understood the meaning, whether he was able to correctly represent the action or process that was described in the task.
How to solve a combinatorial problem?
It is important to correctly determine the type of all connections in the problem under consideration, but it is necessary to check whether there are repetitions of elements in it, whether the elements themselves change, whether their order plays a big role, and also with respect to some otherfactors.
A combinatorial problem can have a number of restrictions that can be placed 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 elements. If there really is an influence, it is necessary to check which one.
Where to start?
First you need to learn how to solve the simplest combinatorial problems. Mastering simple material will allow you to learn to understand more complex tasks. It is recommended that you first start solving problems with restrictions that are not taken into account when considering a simpler option.
It is also recommended to try to solve first those problems in which you need to consider a smaller number of common elements. In this way, you will be able to understand the principle of creating samples and learn how to create them yourself in the future. If the problem for which you need to use combinatorics consists of a combination of several simpler ones, it is recommended to solve it in parts.
Solving combinatorial problems
Such problems may seem easy to solve, but combinatorics is quite difficult to master, some of them have not been solved for the past hundreds of years. One of the most famous problems is to determine the number of magic squares of a special order when the number n is greater than 4.
The combinatorial problem is closely related to the theory of probability, which appeared in medieval times. Probabilitythe 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 get the optimal solution.
Problem solving
Combinatorial problems with a solution are used to teach pupils and students how to work with this material. Generally speaking, they should arouse a person's interest and desire to find a common solution. In addition to mathematical calculations, it is necessary to apply mental stress and use guesswork.
In the process of solving the tasks set, the child will be able to develop his mathematical imagination and combinatorial abilities, this can be seriously useful to him in the future. Gradually, the level of complexity of the tasks to be solved must be increased in order not to forget the existing knowledge and add new ones to them.
Method 1. Bust
Methods for solving combinatorial problems are very different from each other, but all of them can be used by the student to get an answer. One of the simplest, but at the same time, the longest ways is brute force. With it, you just need to go through all the possible solutions without compiling any schemes and tables.
As a rule, the question in such a problem is related to possible variants of the origin of an event, for example: what numbers can be made using the numbers 2, 4, 8, 9? By searching through all the options, an answer is compiled, consisting of possible combinations. This method is great if the number of possible optionsrelatively small.
Method 2. Tree of options
Some combinatorial problems can only be solved by drawing up diagrams that detail information about each element. Drawing up a tree of possible options is another way to find an answer. It is suitable for solving problems that are not too difficult, in which there is an additional condition.
An example of such a task:
What five-digit numbers can be made from the numbers 0, 1, 7, 8? To solve it, you need to build a tree from all possible combinations, while there is an additional condition - the number cannot start from zero. Thus, the answer will consist of all numbers that will begin with 1, 7 or 8
Method 3. Formation of tables
Combinatorial problems can also be solved using tables. They are similar to the tree of possible options, since they offer a visual solution to the situation. To find the correct answer, you need to form a table, and it will be mirrored: horizontal and vertical conditions will be the same.
Possible answers will be obtained at the intersection of columns and rows. In this case, answers at the intersection of a column and a row with the same data will not be obtained, these intersections must be specially marked so as not to get confused when compiling the final answer. This method is not often chosen by students, many prefer a tree with options.
Method 4. Multiplication
There is another way to solve combinatorial problems - the rule of multiplication. He's fineis suitable in the case when, according to the condition, it is not necessary 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 just starting to solve combinatorial problems.
An example of such a task might look like this:
6 people are waiting for the exam in the hallway. How many ways can you use to arrange them in the general list? To get an answer, you need to clarify how many of them can be in the first place, how many in the second, in the third, etc. The answer will be the number 720
Combinatorics and its types
Combinatorial task is not only school material, university students also study it. There are several types of combinatorics in science, and each of them has its own mission. Enumerative combinatorics should consider enumeration and enumeration of possible configurations with additional conditions.
Structural combinatorics is a component of the university program, it studies the theory of matroids and graphs. Extreme combinatorics is also related to university material, and there are individual limitations here. Another section is the Ramsey theory, which deals with the study of structures in random variations of elements. There is also linguistic combinatorics, which deals with the question of the compatibility of certain elements with each other.
Method of teaching combinatorial problems
According to tutorialplans, the age of students, which is designed for primary acquaintance with this material and for solving combinatorial problems, is grade 5. It is there that for the first time this topic is offered for consideration to students, they get acquainted with the phenomenon of combinatoriality and try to solve the tasks assigned to them. At the same time, it is very important that when setting a combinatorial problem, a method is used when the children themselves are looking 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 when solving equations, problems, etc. Thus, combinatoriality plays an important role in further education.
Combinatorial problems: why are they needed?
If you know what combinatorial problems are, then you will not experience any difficulties with their solution. The technique for solving them can be useful when you need to create schedules, work schedules, as well as complex mathematical calculations that are not suitable for electronic devices.
In schools with in-depth study of mathematics and computer science, combinatorial problems are studied additionally, for this special courses, methodological manuals and tasks are compiled. As a rule, several problems of this type can be included in the Unified State Mathematics Exam, 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 problemquickly, since it can have a veiled wording, this is especially important when passing the exam, where every minute counts. Write down separately the information that you see in the text of the problem on a piece of paper, and then try to analyze it in terms of the four ways you know.
If you can put information into a table or other formation, try to solve it. If you can't classify it, in this case it's 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 by solving a certain number of tasks of this type in advance.
Where can I find examples?
The only thing that will help you learn how 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 look for tasks additionally, as a rule, tasks for them are invented by other teachers.
Higher education teachers 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 published repeatedly by the country's leading publishing houses. It is there that you can find tasks that were relevant at that time and remain relevant today.
What if you need to make a combinatorial problem?
Most often, combinatorial problems need to be composedteachers who are obliged to teach students to think outside the box. Here everything will depend on the creative potential of the compiler. It is recommended to pay attention to existing collections and try to compose a problem so that it combines several ways to solve it at once and has different data from the book.
University teachers in this regard are much freer than school teachers, they often give their students the task to come up with combinatorial problems themselves with detailed solution methods and explanations. If you are neither one nor the other, you can ask for help from those who really understand the issue, as well as hire a private tutor. One academic hour is enough to make several similar problems.
Combinatorics - the science of the future?
Many specialists in the field of mathematics and physics believe that it is the combinatorial problem that can become an impetus in the development of all technical sciences. It is enough to take a non-standard approach to solving certain problems, and then it will be possible to answer questions that have been haunting scientists for several centuries. Some of them seriously argue that combinatorics is a help for all modern sciences, especially astronautics. It will be much easier to calculate the flight paths of ships using combinatorial problems, and they will also allow you to determine the exact location of certain celestial bodies.
The implementation of a non-standard approach has long begun in Asian countries, where students evenmultiplication, subtraction, addition and division are solved using combinatorial methods. To the surprise of many European scientists, the technique really works. Schools in Europe have so far only begun to learn from the experience of their colleagues. It is difficult to guess when exactly combinatorics will become one of the main branches of mathematics. Now science is being studied by the world's leading scientists who seek to popularize it.
