This is known as the problem of derangement.
I don't like the notation there, but it gives you the ideas.
I like the formula: .
Well, hi.
I've got a problem, I've been trying to solve it but every formula I try to make fails...
Perhaps this is foolish and I'm not applying correctly the Permutations, Combinations and Factorials...but I don't get to the answer.
The problem is:
Let's Suppose that I'm a teacher, I take some notebooks from nine students, and I give them back. ¿How many ways are there to give back the notebooks in a way that no one receives his own notebook?
What I do, naturally, is to reduce the problem, and try making a formula.
I look what happens with four notebooks...I started looking how many ways there are...and I find there are 9 ways. With 3 notebooks there are only 2 ways. And with 2 there is only 1. No easy-looking pattern, so I can't make a formula that easy...
Then I tried using a formula that I already know...the total quantity of ways to give back the notebooks is n!. Then the formula must be .
When I try to use all of that I get a weird +...-... thing like n!-n!+(n-1)!-...
But it's not exactly the formula...and I'm not sure if it really can be done like that...
This is known as the problem of derangement.
I don't like the notation there, but it gives you the ideas.
I like the formula: .