In general, m (m ≤ n) elements are taken from n different elementsorderArranging into a column is called permutation of m elements from n elements.In particular, when m=n, this arrangement is called all permutation.
Permutation is one of the important concepts in mathematics.Finite setA subset of is arranged in a row, a circle, and no repetition or repetition according to the ordering method of certain conditions.Take m (1 ≤ m ≤ n) different elements out of n different elements each time and arrange them into a column, which is called non repeated arrangement or linear arrangement of m elements out of n elements, which is called arrangement for short.The number of all different permutations of m different elements taken from n different elements is called the number of permutations or the number of permutations, which is recorded as
(or
），
Note: If and only if the two arranged elements are identical and the arrangement order of the elements is the same, the two arrangements are the same.For example,abcAndabdThe elements of are not exactly the same, they are different arrangements;Another exampleabcAndacb, although the elements are identical, they are also arranged in different order.[1]
classification
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There are two kinds of permutation: sorting permutation and full permutation. When m<n, the permutation is called selective permutation in the permutation in which m different elements are taken from n different elements;When m=n, this arrangement is called full arrangement.The number of total permutations of n elements is recorded as Pn，
That is, the permutation number of n different elements taken out is equal to the continuous product of positive integers 1 to n.The continuous product of positive integers from one to n is called nFactorial, use n!express.We stipulate 0=1。
An arrangement of m elements from n elements can be regarded as an ordered subset of m elements of the set A composed of these n elements, so the number of ordered subsets of m elements of A is
。[1]
Formula derivation
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The arrangement formula is to establish a model, take m out of n different elements and arrange them into a column (in order). The first position can have n choices, and the second position can have n-1 choices (one has been placed in the previous position). The same reason can be seen that the third position can have n-2 choices, and so on. The m position can have n-m+1 choices, then the arrangement number
Repeating arrangement(permutation with repetition[2]）It is a special arrangement.M elements can be repeatedly selected from n different elements.Arranging into a column in a certain order is called a repeatable arrangement of m elements from n elements.If and only if the selected elements are the same and the arrangement order of the elements is the same, the two arrangements are the same.
It is easy to know from the principle of fractional counting that the different permutations of repeatable permutations of m elements from n elements are
