But there are three groups of two identical ones, so it needs to be divided by a (2,2) a (2,2) a (2,2) = 8.

So in the end, there is a 720÷8=90 arrangement.

From n different elements, any M (m ≤ n, m and n are natural numbers, the same below) elements are arranged in a column in a certain order, which is called the arrangement of taking out m elements from n different elements; All permutation numbers of m(m≤n) elements taken from n different elements are called permutation numbers of m elements taken from n different elements, which are represented by symbol A(n, m).

Calculation formula:

Besides, the rule is 0! = 1(n！ It means n(n- 1)(n-2)... 1, which is 6! =6x5x4x3x2x 1

Extended data:

Taking out any m(m≤n) elements from N different elements and grouping them is called taking out the combination of M elements from N different elements; The number of all combinations of m(m≤n) elements from n different elements is called the number of combinations of m elements from n different elements. Represented by the symbol C(n, m).

Calculation formula:

; C(n，m)=C(n，n-m).(n≥m)

Other permutation and combination formulas take out the cyclic permutation number of m elements from n elements =A(n, m)/m=n! /m(n-m)！ . N elements are divided into K classes, and the number of each class is n 1, n2, ... nk. The total number of permutations of these n elements is n! /(n 1！ ×n2！ ×...×nk！ ). K-class elements, the number of each class is infinite, and the combined number of M elements extracted from them is C(m+k- 1, m).

Example 1? Arithmetic progression is made up of 1, 2, 3, ... and 20. How many different arithmetic progression?

Analysis: First of all, the complex life background or other mathematical background should be transformed into a clear permutation and combination problem.

Let a, b and c be equal,

∴ 2b=a+c, which means that B is determined by A and C,

And ∵ 2b is an even number, ∴ a and c are also odd or even numbers, that is, choose two numbers from the ten numbers 1, 3, 5, ..., 19 or 2, 4, 6, 8, ..., 20, so that arithmetic progression, a (.

References:

Baidu encyclopedia-permutation and combination