mathematics teachers introduce their students to the concept of "combinatorial problem" is still in the fifth grade.This is necessary to ensure that they were able to continue working with more complex tasks.Under the combinatorial problem can be understood as an opportunity to resolve it with the brute force of a finite set.
main feature of problems of this order is the question to them, which sounds like "What options?" Or "How many ways?" Combinatorial problems depends on whether understood if the crucial their meaning, managed if it is properly represent the action or process thatIt has been described in the job.
How to solve the combinatorial problem?
It is important to correctly determine the type of all available in the problem compounds, but it is necessary to check as to whether it repeats elements to see if the elements themselves whether to play a greater role of the procedure, as well as on several other factors.
combinatorial problem may have a number of limitations that may be imposed on the connection.In this case, you will need to count all her decision and to verify whether these restrictions have any influence on the connection of all components.If the effect is really there, you must check what kind.
Where to start?
First we need to learn how to solve simple combinatorial problems.Mastering simple materials will allow to learn to understand more complex tasks.It is recommended to first start to solve the problem with the restrictions that are not taken into account when considering a simpler version.
also encouraged to try to solve those problems first, which you should consider a smaller number of common elements.So you can understand the principle of creating samples and learn how to create them on their own future.If the task for which you want to use a combinatorics, consists of a combination of several more simple, it is recommended to solve it in parts.
combinatorial problems
These tasks may seem simple in the decision, but the combinatorics rather complicated to master, some of them do not have solutions for the past hundred years.One of the most prominent problems is the determination of the number of special-order magic squares, when the number n is greater than 4.
combinatorial problem is closely related to the theory of probability, which appeared in medieval times.Chance of origin of an event can only be calculated using combinatorics, in this case, you will need to alternate places of all the factors to get the best solution.
Problem Solving
combinatorial problem with a solution used to teach pupils and students working with this material.If we talk in general, they should make a person of interest and a desire to find a common solution.In addition to mathematical calculations, it is necessary to apply the mental stress and use a guess.
In the process of solving the tasks the child can develop their imagination and combinatorial mathematical ability, it can seriously be useful to him in the future.Gradually, the level of complexity of the tasks necessary to improve, not to forget the existing knowledge and add to them new.
Method 1. Bust
Methods for solving combinatorial problems are very different from each other, but they can be used for pupil response.One of the most simple, but at the same time and the longest way is overkill.When it is necessary to simply try all possible solutions without making any diagrams and tables.
As a rule, the issue in such a problem associated with possible origin of a particular event, such as what numbers you can make using the digits 2, 4, 8, 9?By iterating through all the options drawn up a response consisting of possible combinations.Such a method is perfectly fine if the number of options is relatively small.
Method 2: Tree option
Some combinatorial problems can be solved only by making the scheme, which will detail contains the information about each item.Making wood options - another way to find the answer.It is suitable for solutions not too difficult tasks in which there is an additional condition.
example of such a problem:
- What are five-digit numbers can be formed from the digits 0, 1, 7, 8?To solve the need to build the tree of all possible combinations, while there is an additional condition - the number can not start with zero.Thus, the response will consist of all the numbers that will start with 1, 7, or 8.
Formation Method 3 tables
combinatorial problems can be performed with the help of tables.They are similar to the tree of options, as it offers a clear solution to the situation.To find the correct answer should be to form a table, and it will be mirrored: horizontal and vertical conditions are the same.
possible answers will be obtained at the intersection of columns and rows.This answers to the intersection of the row and column with the same data will not be obtained, the intersection should be particularly mark to avoid confusion in the preparation of the final answer.This method is not too often chosen disciples, many prefer a tree with options.
method 4. Multiply
There is another way by which you can solve combinatorial problems - multiplication rule.It is perfect in the case when the condition is not necessary to list all the possible solutions, you just need to find the maximum number.This method is unique, it is used very often, when only begin to solve combinatorial problems.
example of this problem may look like this:
- 6 people expect in the exam hall.How many ways can be used to place them in the list?To get the answer you need to specify how many of them can be in the first place, but on the second, third, and so on. D. The answer will be the number 720.
Combinatorics and its species
combinatorial problem is not the only school materials, university students are also studying it.In science, there are several types of combinatorics, and each of them has its own mission.Combinatorial enumeration should consider the problem to transfer and counting of possible configurations with additional conditions.
Structural combinatorics is a component of the high school program, it examines the theory of matroids and graphs.Extreme combinatorics also has to do with the material to high school, and here are their individual limitations.Another section - Ramsey theory, which studies patterns in random variations of elements.There is also a linguistic combinatorics, which is considering the compatibility of certain elements among themselves.
Teaching method of combinatorial problems
according to the curriculum, the age of the students, which is designed for initial acquaintance with the material and combinatorial problems - 5th grade.It was there for the first time this topic is offered to the students, they get acquainted with the phenomenon of combinatorial and try to solve their tasks.It is very important that the formulation of the problem of combinatorial method was used when children are engaged in the search for answers to questions.
In addition, after studying this topic would be much easier to introduce the concept of factorial and use it for solving equations, tasks and so on. Thus, combinatorial plays an important role in further education.
Combinatorial problems: what are they for?
If you know what a combinatorial problems, no difficulties with their decision you will experience.Methods of solving them can be useful, if necessary, scheduling, work schedules and complex mathematical calculations to perform are not suitable electronic devices.
In schools with in-depth study of mathematics and computer science combinatorial problems are studied further, for this is a special courses, manuals, and tasks.As a rule, several problems of this type can be part of the unified state examination in mathematics, they are usually "hidden" in Part C.
How to solve the combinatorial problem quickly?
It is important to discern the combinatorial problem quickly, since it may be a veiled wording, this is particularly important at the time the exam, where every minute counts.Write out separately the information that you see in the text of the problem on a sheet, and then try to analyze it from the point of view of the four famous ways.
If you can put information into a spreadsheet or other entity, try to solve it.If we classify it, you can not, in this case it is best to leave it for long and move on to other tasks, so as not to lose valuable time.This situation can be avoided in advance poreshat a number of problems of this type.
Where can I find examples?
only thing that will help you learn how to solve combinatorial problems - examples.They can be found in special mathematical collections, which are sold in the shops of educational literature.However, there can only find information for the students of the university, students will have to find the problem further, as a rule, they invented the rest of the job teachers.
University professors believe that students need to train constantly and offer them additional educational literature.One of the best collections considered "Methods of Discrete Analysis in solving combinatorial problems", written in 1977 and discharged repeatedly by leading publishing houses of the country.That's where you can find the tasks that were relevant at the time and remain valid today.
What to do if you want to make a combinatorial problem?
Most often combinatorial problems need to be teachers who are required to teach students to think unconventionally.Here everything will depend on the creativity of the originator.It is recommended to pay attention to the existing collections and try to make the problem so that it combines several ways to solve it, and was different from the book data.
University professors in this plan is much freer school, they often give their students to come up with the task by the combinatorial problems, with detailed explanations of the methods and solutions.If you are neither the one nor the other, you can ask for help from those who really know the area, as well as to hire a private tutor.One academic hour is enough to create several similar tasks.
Combinatorics - the science of the future?
Many experts in the field of mathematics and physics believe that it is combinatorial problem could trigger the development of technical sciences.Suffice it to unconventional approach to solving various problems, and then we can answer the questions that have already been several centuries haunt scientist.Some of them are seriously argue that combinatorics is a tool for all modern science, especially space exploration.It is much easier to calculate the trajectory of the ships using the combinatorial problems, as they will determine the exact location of certain celestial bodies.
Implementation of non-standard approach has long begun in Asian countries, where students even basic tasks of multiplication, subtraction, addition and division decide, using combinatorial methods.To the surprise of many European scientists, the technique really works.Schools in Europe so far only begun to learn from the experience of their colleagues.When it combinatorics will be one of the major branches of mathematics, to assume difficult.Now science is studied the planet's leading scientists who seek to popularize it.
