What are all the ways to permutate and combine? How do they do it? Columns such as interpolation 40

Permutations and combinationsInterpolationExample questions:

There are 10 street lights numbered 1 to 10 on the side of the road, and now you want to turn off 3 of them, but you can't turn off the adjacent 2 or 3 lights, and you can't turn off the street lights at both ends, how many ways to turn off the lights that meet the requirements?

Solution: (interpolation).

This problem is equivalent to inserting 3 lights out of 6 gaps between 7 on-on street lights, so the total number of methods sought is c(6,3)=20 methods, which cannot be represented by a, because it is a combination problem.

Two commonly used permutations: basic counting principles and applications:

1. Addition principle and categorical counting method:

Each method in each category can accomplish this task independently, and the specific methods in the two different types of methods are different from each other (i.e., the classification is not duplicated), and any method to complete this task belongs to a certain category (i.e., the classification is not omitted).

2. Multiplication principle and step-by-step counting method:

One method of any step cannot complete this task, and this task can only be completed by completing these n steps consecutively, and each step count is independent of each other, as long as the method taken in one step is different, the corresponding method of completing this task is also different.

Formula: c(n,m)=a(n,m) a(n,n) Explain the principle of elimination from the above formula.

a(n,m) is an arrangement of n elements taken from m elements, and the same elements are arranged differently due to different orders.

c(n,m) is a combination of n elements taken from the m element, and since the order is not considered, the same elements can only form a combination. Each combination corresponds to a (n,n) permutation, c(n,m) = a(n,m) a(n,n) (de-ordered).

Coefficient Properties:and the coefficient of equal distance between the first and last ends;

When the binomial exponent n is odd, the middle two terms are largest and equal;

When the binomial exponent n is even, the middle term is maximum;

In a binomial, the odd and even terms are the same sum and are both 2 (n-1);

The sum of all coefficients in binomial is 2 n

The above content reference: Encyclopedia - Permutations and Combinations.

6 kinds. 1. Here are the permutation problems in mathematics, which can be enumerated by step-by-step discussion:

2. The first position is a triangle, and such a combination of forms are: triangle, square, circle or triangle, circle, square.

3. The first position is a square, and the forms of such a group chain are: square, circle, triangle or square, triangle, circle.

4. The first position is circular, and such a combination of forms are: circle, triangle, square or circle, square, triangle.

5. There are a total of 6 forms of all permutations and combinations.

There are 5 combinations, and the solution in high school math is c[5,4]=5.

The elementary school math solution is to divide 5 numbers into 2 groups, the first group has 4 numbers, and the second group has 1 number, that is to say, when the 1 number of the second group is determined, the first group of numbers is determined. Since there are 5 combinations for the second group, there are also 5 combinations for the first group.

The arrangement is related to the order of the elements, and the combination has nothing to do with the order For example, 231 and 213 are two permutations, and the sum of 2 3 1 and 2 1 3 is a combination

a) Two basic principles are the basis for permutations and combinations.

1) Addition principle: to do one thing, to complete it can have n types of methods, in the first type of methods there are m1 different methods, in the second type of methods there are m2 different methods ,......There are mn different approaches in the nth category, so there are n m1 m2 m3 to accomplish this....mn different methods

2) Multiplication principle: to do one thing, to complete it needs to be divided into n steps, there are m1 different ways to do the first step, and m2 different ways to do the second step,......There are mn different ways to do step n, so there are n m1 m2 m3 to complete this thing....mn different methods

Here it is necessary to pay attention to distinguish between the two principles, to do one thing, if there are n types of methods to complete it, it is a classification problem, and the methods in the first category are all independent, so the principle of addition is used; To do a thing, it needs to be divided into n steps, and the steps are continuous between the steps, and only a number of interrelated steps that will be divided into them are completed in turn, so the principle of multiplication is used

There is an essential difference between the "class" and the "step" of accomplishing a thing in this way, so it also distinguishes the two principles

b) Permutations and number of permutations.

1) Arrangement: From n different elements, take any m (m n) elements and arrange them in a column in a certain order, which is called taking out an arrangement of m elements from n different elements

From the meaning of the arrangement, we can see that if the two permutations are identical, not only the elements of the two permutations must be exactly the same, but also the order of the permutations must be exactly the same, which tells us how to judge whether the two permutations are the same

2) Permutation number formula.

Take out all permutations of m(m n) elements from n different elements.

When m n, it is the full permutation pnn=n(n 1)(n 1)....3·2·1＝n！

iii) Combinations and number of combinations.

1) Combination: From n different elements, take any m (m n) elements and form a group, which is called a combination of m elements from n different elements

From the definition of combination, if the elements in two combinations are exactly the same, they are the same combination regardless of the order of the elements; Only when the elements in two combinations are not exactly the same are they different combinations

2) Number of combinations: take out all combinations of m(m n) elements from n different elements.

Here we should pay attention to the difference and connection between arrangement and combination, from n different elements, any m (m n) elements, "in a certain order in a column" and "in any order and in a group" is essentially different

Your siblings are the arrangement, and the parents are the combination.