COMBINATORY PROBLEMS AND THEIR SOLUTION METHODS

Аннотация

this article reveals the features of teaching students methods of solving problems in combinatorics when studying a school course in mathematics, and also considers methods for solving historical combinatorial problems, problems on the multiplication rule, developing skills in solving problems on the formation of combinatorial concepts, tree of options, factorial, application to solving equations and simplifying expressions. When discussing formulas for figured numbers, it is proposed to give students tasks to independently find a pattern and verify their correctness for specific values of a natural variable, and then solve problems using a formula to find a certain number by location and type of number; inverse problems by number to find what type a given number belongs to, when discussing solving problems similar to the well-known problem, in which it is required to find the number of possible seating positions by people, the number of which is equal to the number of seats, it is indicated to first number the seats, and then find the number of permutations of seating of these seats by people. Keywords: combinatorics, permutations, placement, combinations, tasks, multiplication rule, skills, thinking, development, skills, concepts, tree of options. Introduction. Each combinatorial problem is different, and there is no single method suitable for any of them. You always have to think and look for solutions. At the same time, many problems, although different in content, are similar in the approach to solving them and allow the use of similar reasoning techniques. This makes it possible to pay attention to a number of typical situations, to give a certain set of “keys” that can be tried in order to “unlock” the problem. Although, of course, there is no guarantee of success: for another non-standard task, not a single key from the existing set may be suitable, and a new one will have to be “made.” Tasks can be divided into elementary and combined. Elementary ones are those that are solved in one action; To solve them, it is enough to select the appropriate formula and carry out calculations. In these problems, the most important thing is to analyze the condition and correctly determine 44 ISSN (E): 2181-4570 ResearchBib Impact Factor: 6,4 / 2023 SJIF 2024 = 5.073/Volume-2, Issue-7 the type of combinations [2]. In combined problems of placement, permutation, and combinations turn out to be components of more complex combinations. The usual approach to combination problems is to break them down into elementary ones, find the answer in each such part, and then look for a way to count those more complex combinations that meet the conditions of the problem. The principle of sum and product. At the final stage of solving combinatorial problems, the following situations often arise: - to implement a complex combination, each of the elementary situations that form it must take place; - to carry out a complex combination, it is enough that at least one of the elementary combinations that form it takes place. Let's consider an example, which at the same time will allow us to detect patterns that arise in each of the noted situations. In the first case, we reason like this: since the order of choice does not matter, we assume that we choose the apple first. This can be done in three ways. For each of the selected apples, you can attach any of the two pears. Each choice of an apple leads to the formation of 2 “apple-pear” pairs, and there are 3 such choices, which means that there will be all ways to choose an apple and a pear. If we started by choosing a pear, we would end up with ways to choose a pear and an apple, i.e. the same result. Answer to the question “In how many ways can you choose an apple and a pear?” we got it this way: we found separately the number of choices for an apple and the number of choices for a pear, and since the question required choosing an apple and a pear, we multiplied the found numbers. In the second case, we start with the fact that an apple can be selected in three ways, and a pear in two. But now we need to choose an apple or a pear, that is, one fruit. It is clear that there will be ways to do this. Thus, to determine the number of choices of an apple or a pear, the number of choices of an apple and the number of choices of a pear should be added. It is necessary to pay attention: during the final calculation of the number of combinations corresponding to the conditions of the problem, the logical connective “and” led to finding the answer by multiplication, and the logical connective “or” - by addition. Product principle: if combination A can be accomplished in ways, and combination B (regardless of A) in ways, then both combinations together can be accomplished in ways. Indeed, in each of the ways of implementing combination A, 45 ISSN (E): 2181-4570 ResearchBib Impact Factor: 6,4 / 2023 SJIF 2024 = 5.073/Volume-2, Issue-7 pairs are formed consisting of combinations A and B. This mea