definition:
P (m,n): Take the number of permutations of m elements from n different elements.
C(m,n): Take the number of combinations of m elements from n different elements
official:
P (n,m)=n(n-1)(n-2)…(n-m+1)
Example: P(2,5)=5 * 4=20
C（n,m）=n(n-1)(n-2)…(n-m+1)/m!
Example: C(2,5)=5 * 4 /1 * 2=10
Circle arrangement
Take out m(1<=m<=n) elements from n different elements without repeating them and arrange them on the circle, find out how many different arrangement methods there are in total
(if one m-circle is arranged and rotated, another m- Circle arrangement, the two circles are considered the same arrangement)
Formula: P(m,n)/m
Misalignment
There is an arrangement of n elements. If all elements are not in their original positions after the rearrangement, then this arrangement is called a stagger of the original arrangement. How many different staggering methods are there
Formula: D(n)=(n-1) [D(n-1) + D(n-2)]
In particular, D(1) = 0, D(2) = 1
Special conditions for arrangement
Example: 7 students stand in a row, how many different arrangements are there according to the following requirements
①A certain fixed position of station A:
P(6,6)
②A stands in the middle, and B is adjacent to A:
P（2,2） P（5,5）
③A and B are adjacent:
P（2,2）P（6,6）
④A and B cannot be adjacent:
P（7,7）-P（2,2）P（6,6）
⑤ A, B, and C are adjacent to each other:
== P(5,5)P(3,3)==
⑥The two persons of A and B do not stand at the head and end of the row:
P（2,5）P（5,5）
⑦A, B, and C are not adjacent to any two:
P（3,5）P（4,4）
⑧ A and B must be adjacent to each other, and C does not stand at the head or end of the row:
P（2,2）P（1,4）P（5,5）
Combination of n balls and m boxes
N balls, m boxes, whether the balls are the same, whether the boxes are the same, whether the boxes can be empty, a total of 2^3=8 types (only the common two are discussed below):
1. Same ball, different box, no empty box
C(n-1,m-1)
2. Same ball but different box, empty box allowed
C(n+m-1,m-1)