Position analysis and element analysis are the most common and basic methods to solve the problem of permutation and combination. If element analysis is the main method, special elements should be arranged first, and then other elements should be processed. If position analysis is the main method, it is necessary to meet the requirements of special positions before dealing with other positions; If there are multiple constraints, it is usually necessary to consider one constraint in addition to other conditions.

This problem of determining whether to arrange or combine first does not need to consider the order of the first and last digits, but the first and last digits have priority requirements, so the first and last digits must be arranged first, and the last digits must be an odd number, that is, 1, 3 and 5, and the first digit cannot be arranged as 0. After excluding an odd number, there are only four digits to choose from, and we can arrange the remaining three digits directly.

The second category: the combination strategy of adjacent/interphase elements.

The problem that several elements must be arranged together can be solved by binding, that is, adjacent elements need to be merged into one element and then arranged with other elements, and attention should be paid to the arrangement within the merged elements. Be sure to pay attention to keywords when reviewing questions.

Type III. Interpolation strategy for nonadjacent problems

Firstly, the elements without position requirements are queued, and then non-adjacent elements are inserted in the middle and at both ends.

Therefore, the keywords of these two methods are adjacent. If the adjacent elements are attached, they should be regarded as a whole, that is, the "binding method" should be adopted. If some elements cannot be adjacent as an additional condition, interpolation method can be adopted. "Insert" includes "insert" at the same time and "insert" one by one. Pay attention to the restrictions of conditions.

Type four. Double contraction vacancy insertion strategy for scheduling problem]

"Division" is used for fixed-order problems. For the problem that some elements are arranged in a certain order, we can first arrange these elements with other elements, and then divide the total arrangement number by the total arrangement number of these elements. Of course, you can also use the double contraction method, and you can also convert it into an interpolation model that takes up space.

Type 5: Power search strategy for rearrangement problem.

Housing allocation is also called "Hotel Law" and the power strategy of rearrangement. In order to solve the problem of allowing repeated arrangement, we should pay attention to distinguish two types of elements: one type of elements can be repeated and the other type of elements cannot be repeated. We should regard the unrepeatable elements as "guests" and the repeatable elements as "shops", and then directly solve them by the principle of multiplication. The characteristic of the permutation problem that allows repetition is that the elements are taken as the research object, and the elements are not limited by their positions, so the positions of each element can be arranged one by one. Usually, the arrangement number of n different elements in m positions is mn.

Example: How many different ways are there to divide six interns into seven workshops for internship?

Type 6, cyclic arrangement problem

Type 7, multi-line problem

Generally speaking, the arrangement of elements divided into multiple lines can be reduced to one line and then studied in sections.

Type 8, small group problem

In the arrangement of small groups, the whole should be followed by the part, and then other strategies should be combined to deal with it.

Type 9, partition strategy for problems with the same elements.

Type 10. If it is difficult, it is to oppose the overall elimination problem.

For some complex or abstract permutation and combination problems, we can use the idea of transformation to simplify them into simple and specific problems to solve. For some permutation and combination problems, the direct consideration of the positive side is more complicated, while its negative side is often simpler. We can find its negative side first and then eliminate it from the whole. For problems with negative words, we can also subtract the unqualified ones from the whole. At this time, we should pay attention to neither more nor less.

Type 1 1, average block partition problem

12, practical operation enumeration problem

Type thirteen. Analyze specific issues

To solve the problem of constrained permutation and combination, we can classify the elements according to their nature and step by step according to the continuous process of events, so that the standard is clear. The step-by-step level is clear, and once the classification standard is determined, it will run through the process of solving problems. When dealing with complex permutation and combination problems, we can degenerate a problem into a short problem, and find a solution by solving this short problem, so as to solve the original problem in the next step.

abstract

Although the model of permutation and combination is changeable, in fact, what teachers like best is concrete analysis of specific problems. According to the most basic principle of addition principle sum multiplication, the problem of permutation and combination is solved, and the complexity is simplified. In the remaining 20 days of the college entrance examination, we should think more about the breakthrough point of the exam questions, not just the conclusion, but the understanding. What we should really think about is, if I don't have the answer, how to start next time I encounter this type of problem, how to solve it, and how to ensure the score. I hope these skills summarized are helpful to everyone, and it is not difficult to arrange and combine them. Come on!