Current location - Training Enrollment Network - Mathematics courses - In order to celebrate New Year's Day, each of the four students in the dormitory prepared a gift as an exchange. It is stipulated that if you don't bring your own gift, how many ways can you have?
In order to celebrate New Year's Day, each of the four students in the dormitory prepared a gift as an exchange. It is stipulated that if you don't bring your own gift, how many ways can you have?
Concept: all the dislocated arrangements, n kinds of substances, are rearranged so that they are not in their original positions.

History: Two special cases of the "error envelope problem" called "a wonderful problem of combinatorial number theory" by the famous mathematician Euler (1707- 1783).

The problem of "misplaced envelopes" was raised by Daniel Bernoulli, the son of the most famous mathematician at that time (1700-1782). The main ideas are as follows:

A person wrote n different letters, corresponding to n different envelopes. He put all these n letters in the wrong envelope. How many ways can you put them in the wrong envelope?

Formula:

Recursive formula: Let the arrangement of N substances be s(n).

When n= 1, s(n)=0.

When n=2, s(n)= 1.

Otherwise s (n) = (n-1) * (s (n-1)+s (n-2))

Factorial formula: n! -C( 1 n)*(n- 1)! +C(2 n)! -…+(- 1)^kC(k n)*(n-k)! +…+(- 1)^n*C(n n)*0!

The above formula is equivalent to n! ( 1- 1/ 1! + 1/2! - 1/3! +...+(- 1)^n* 1/n! )

________________________

For this question:

S( 1)=0,S(2)= 1

S(3)=2(0+ 1)=2

S(4)=3(2+ 1)=9

If the number of questions is small, it can also be listed directly: suppose the order of the four students is (A, B, C, D),

(B,A,D,C) (B,C,D,A) (B,D,A,C)

(C,A,D,B) (C,D,A,B) (C,D,B,A)

(D,A,B,C) (D,C,A,B) (D,C,B,A)

The above 9 kinds.