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！ )

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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.