Teachers of mathematics introduce their students to the concept of “combinatorial problem” in the fifth grade. This is necessary in order for them to be able to work with more complex tasks in the future. By combinatoriality of a problem, one can understand the possibility of solving it by enumerating elements of a finite set.
The main sign of tasks of this order is the question for them, which sounds like "How many options?" or "How many ways?" The solution of combinatorial problems directly depends on whether he understood the decisive meaning, whether he managed to correctly represent the action or process that were described in the task.
How to solve a combinatorial problem?
It is important to correctly determine the type of all the compounds available in the problem under consideration, but it is necessary to check whether there are repeats of elements in it, whether the elements themselves change, whether their order plays a large role, and also regarding some other factors.
A combinatorial task can have a number of constraints that can be imposed on compounds. In this case, it will be necessary to fully calculate its solution 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 begin?
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 that you first try to solve those problems in which you need to consider fewer common elements. Thus, you can understand the principle of creating samples and learn how to create them yourself in the future. 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.
The solution of combinatorial problems
Such tasks may seem easy to solve, but combinatorics are difficult enough to master, some of them have not been solved for the past hundreds of years. One of the most famous tasks is to determine the number of special-order magic squares when the number n is greater than 4.
The combinatorial problem is closely related to the theory of probability, which appeared in medieval times. The probability of the occurrence of a particular event can only be calculated using combinatorics, in this case it will be necessary to alternate all the factors in order 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 interest and desire in a person to find a common solution. In addition to mathematical calculations, it is necessary to apply mental stress and use the hunch.
In the process of solving the set tasks, the child will be able to develop mathematical imagination and combinatorial abilities, this can be useful to him in the future. Gradually, the difficulty level of the tasks to be solved must be increased so as not to forget the existing knowledge and add new ones to them.
Method 1. Search
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, and the longest ways is brute force. With him, you just need to sort through all the possible solutions, without making any schemes and tables.
As a rule, the question in such a problem is connected with possible variants of the origin of a particular event, for example: what numbers can be composed using the numbers 2, 4, 8, 9? By enumerating all the options, an answer is made up of possible combinations. This method is perfect if the number of possible options is relatively small.
Method 2. Tree of options
Some combinatorial problems can be solved only by drawing up schemes in which information about each element will be indicated in detail. Compiling a tree of options is another way to find the answer. It is suitable for solving not too complicated problems in which there is an additional condition.
An example of such a task:
- What five-digit numbers can be made up of the numbers 0, 1, 7, 8? To solve this, you need to build a tree from all possible combinations, and 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, because they offer a visual solution to the situation. To find the right 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 rows. At the same time, answers at the intersection of a column and a row with the same data will not be received, these intersections must be specially marked so as not to get confused when drawing up the final answer. This method is not very often chosen by students, many prefer the tree with options.
Method 4. Multiplication
There is another way with which you can solve combinatorial problems - the rule of multiplication. It is perfect 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 just begin to solve 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 put them on the general list? To get an answer, it is necessary 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
The 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. Enumerator combinatorics should consider the tasks of enumerating and counting possible configurations with additional conditions.
Structural combinatorics is a component of the university curriculum, it studies the theory of matroids and graphs. Extreme combinatorics is also related to university material, and there are individual limitations. Another section is Ramsey's theory, which studies 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.
Methods of teaching combinatorial problems
According to the curriculum, the age of students, which is designed for an initial acquaintance with this material and for solving combinatorial problems, is grade 5. It was there that for the first time this topic was proposed for consideration by students, they get acquainted with the phenomenon of combinatorialism and try to solve the tasks assigned to them. Moreover, it is very important that when setting the 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 to solve 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 solution. The technique for solving them can be useful if you need to draw up schedules, work schedules, as well as complex mathematical calculations, for which electronic devices will not work.
In schools with in-depth study of mathematics and computer science, combinatorial problems are studied additionally, for this purpose special courses, teaching aids and tasks are compiled. 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 make out a combinatorial task quickly, 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 task on a piece of paper, and then try to analyze it from the point of view of four methods known to you.
If you can put information in a table or other education, try to solve it. If you cannot classify it, in this case it is best to leave it for a short while and move on to solving another problem 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 things that help you learn how to solve combinatorial problems are 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, students will have to look for tasks in addition, as a rule, for them tasks are thought up by other teachers.
University professors believe that students need to train and constantly offer them additional educational literature. One of the best collections is considered “Methods of discrete analysis in solving combinatorial problems”, written in 1977 and published repeatedly by leading publishers of the country. It is there that you can find tasks that were relevant at that time and remain relevant today.
What to do if you need to compose a combinatorial problem?
Most often, combinatorial tasks need to be made up by teachers who are required to teach students to think outside the box. Everything here 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 for solving it at once and has other than book data.
In this regard, university teachers are much freer than school teachers; they often give their students the task of themselves to come up with combinatorial problems with detailed solution methods and explanations. If you do not belong to either one or the other, you can ask for help from those who are really versed in the matter, and also hire a private tutor. One academic hour is enough to compose several similar tasks.
Combinatorics is the science of the future?
Many experts in the field of mathematics and physics believe that it is a combinatorial problem that can become an impetus in the development of all technical sciences. It is enough to approach non-standard approach to the solution of various 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 an aid to all modern sciences, especially astronautics. It will be much easier to calculate the flight paths of ships using combinatorial tasks, they will also allow you to determine the exact location of various celestial bodies.
The implementation of a non-standard approach has long begun in Asian countries, where students even solve elementary problems of multiplication, subtraction, addition and division using combinatorial methods. To the surprise of many European scientists, the technique really works. Schools in Europe have just begun to learn from the experience of their colleagues. When exactly combinatorics will become one of the main branches of mathematics, it is difficult to assume. Now science is being studied by leading scientists of the planet who seek to popularize it.