1, counting principle:

(1) classification counting principle (addition principle);

There are n ways to accomplish a thing, the first way has different ways, the second way has different ways, …… The nth way has different ways, so the way to accomplish it is also different.

(2) Step-by-step counting principle (multiplication principle):

To accomplish one thing, it needs to be divided into n steps. There are different ways to do the first step, the second step, the N step and the N step.

(3) The difference between the two principles is that one is related to classification and the other is related to gradual progress, that is, "mutual exclusion":

(1) has the following three points for addition principle:

I "exclusion"-mutually exclusive independent events;

Mode Ⅱ: "Doing things"-"Classification"-"Addition";

Iii. Key points: Grasp the standard of classification, classify properly, and ensure that classification is not omitted or repeated.

(2) The multiplication principle has the following three points:

I "connect" related events;

Mode Ⅱ: "Doing things"-"Step by Step"-"Multiplication";

Ⅲ Key: Grasp the characteristics step by step, and design the step-by-step scheme correctly, so that each step is interrelated and independent.

Second, review the finishing touch:

1, classified counting principle and step-by-step counting principle are the basic principles of counting problems, which embody two common methods of decomposing problems, namely, classified solving and step-by-step solving.

(1) The function of two basic principles: calculating all the different ways to do one thing and finish it;

(2) The difference between two basic principles: one is related to classification and the other is related to gradual progress; Addition principle is "classified completion" and the principle of multiplication is "step by step completion";

(3) Two principles:

(1) In the principle of classified counting (addition principle), "there are n ways to accomplish one thing" means that each method is mutually exclusive, that is, each method can be accomplished independently, and at the same time, they are independent and do not interfere with each other. The intersection is an empty set, and the set is a complete set, without repetition or omission. When classifying, all methods are required to be mutually exclusive. Either way, it can be done independently. Only when this condition is met can addition principle be directly used, otherwise it can't be done.

(2) In the step-by-step counting principle (multiplication principle), "to complete a thing, it needs to be divided into n steps", which means that each step is not enough to complete, and these steps cannot be repeated or omitted. If a thing needs to be divided into several steps, each step is essential, and all the steps need to be completed, and the steps are independent of each other, that is, what method is used in the previous step does not affect what method is used in the next step, and there are m different methods in the next step, then the number of methods to complete it can be directly used by the multiplication principle.

(4) Similarities and differences between the two principles:

1 * * * Similarity: It's all counting principles, that is, counting principles of different ways to accomplish one thing, so we must first find out what it is and how to accomplish it;

② Difference: The N methods in the principle of classification and counting are independent of each other, and each method in each category can accomplish this independently; Each step in the step-by-step counting principle is interdependent, and each step cannot accomplish things independently. Only when all the steps have been completed can this thing be completed.

(5) Steps to solve problems with two principles:

Step 1: specify what to accomplish, and determine whether to divide the event into "N classes" or "N steps" according to the characteristics of the event;

Step 2: Find out the number of different methods in each class or step;

Step 3: Use "addition" or "multiplication" to get the total number of methods to complete the event;

Step 4: Answer.

2. When solving the counting problem with two counting principles, the most important thing is to analyze carefully before starting the calculation to determine whether it is necessary to classify or step by step:

(1) The classification should be "neither weight nor leakage"-no leakage. After classification, each category is counted separately, and finally the sum is obtained by using the principle of classification addition counting.

(2) Step by step to achieve "all steps"-be short of one, complete all steps, just complete the task, of course, "steps" and "steps" should be independent of each other, but also pay attention to the continuity between "steps". After step by step calculation, the number of methods in each step. Finally, according to the principle of step-by-step multiplication and counting, the number of completed methods in each step is multiplied to get the total.

3. When solving the problem of counting principle, we should start with simple classification discussion, try to solve more than one problem and consider the problem from different angles. For complex problems, we can use two basic counting principles at the same time or use the method of list and circle to help analyze.