On the issue of permutations and combinations of misalignments

This is called the all-wrong permutation problem, and it was first answered by Euler. We might as well take n people as f(n), then f(n)=(n-1)[f(n-1)+f(n-2)].f(0)=0,f(1)=1.

This recursive formula is easy to prove.

The proof is as follows: Let n people be a, b, c, d....n cards are A, B, C, D....

If A takes B's card B, and B also takes A's card A, then it is obvious that only n-2 people are left to take the card, and it is naturally F(n-2) species.

If A takes B's card B, B doesn't take A's card A (with"B didn't take B's card B"The same), then it is obviously the same as n-1 people taking cards, naturally it is f(n-1) species.

And A doesn't necessarily take B, as long as it's B, C, D....n-1) is fine, so multiply f(n-1) + f(n-2) by n-1.

If you have learned to solve an abstract function equation, the solution of f(n)=(n-1)[f(n-1)+f(n-2)] in a natural number is f(n)=n![1/2!-1/3!

1)^n/n!(f(n)=1 at n=1).

The abstract function is correct, and the recursive formula is correct, but it should be noted that in the actual case f(1)=0, the assumption that f(1)=1 is problematic.

My understanding is that f(1)=0, f(2)=1, and for f(n) you should start using the formula from n greater than or equal to 3.

If f(1) 1, then according to the recursive formula, f(2) 1 (0+1) 1, f(3) 2 (1+1) 4, this is wrong, in fact f(3) should be equal to 2. (Based on the abstract functions listed above, f(3) 6 (1 2-1 6)=2.) ）

Find the total number n first, and then subtract the number n that does not take yourself

1. All because of the question by default, these 20 tasks are the same =. =

If the 20 tasks are different, the answer will be wrong, even if you divide the two tasks and do not arrange them, the interpolation method of the answers is also wrong, and you will also notice the two tasks.

But if these 20 tasks are the same, then the answer makes sense.

3 4 (3 to the 4th power) means that each tiger has 3 choices to be scattered, but the pure acorn requirement is that "the same sheep cannot be eaten by different tigers", so assuming that after Tiger A eats A sheep, then the BCD tiger cannot eat A sheep, so the further you go, the smaller the tiger's choice, so it can't be 3 4

I think it should be understood in this way, because there are more tigers than sheep, so it should be the other way around, and let the sheep decide which tiger to eat, so I think the answer should be 4x3x2=12.

After the answer, if there is anything wrong, welcome to communicate and learn to dig out!

Each sheep can be eaten by one of the four tigers, so there are four possibilities, i.e. 4*4*4=64 species.

There are six ways to choose type A, and there are four kinds of wheel cavities corresponding to each method B of the old shirt, so it is....

3. I don't understand Tong Qiao.

Column** is it!