1. ## Combinatory problem

Suppose there are N balls labeled 1, 2, 3, ... , N, and N boxes also labeled 1, 2, 3, ..., N.
If you put one ball inside of each box, how many combinations are possible so that there are no match of the ball label and box label (ball 1 is not in box 1, ball 2 is not in box 2, and so on)?

2. Hello, lcmarincek!

Is this a homework problem?

If so, several questions arise:

. . First of all, who assigned it?
. . What course are you in?
. . What is your mathematical background?

. . How "good" is your teacher?
. . Would he/she recognize the right answer if shown it?

This is a complex problem with a highly elusive solution.
I'd prefer not to explain it ... if it goes pffft! over your heads.

Suppose there are $N$ balls labeled 1, 2, 3, ... , N
and $N$ boxes also labeled 1, 2, 3, ..., N.
If you put one ball inside of each box, how many combinations are possible
so that there are no match of the ball label and box label?
Such an arrangement is called a "derangement of $N$ objects."

I'll let someone else explain it.

3. Originally Posted by lcmarincek
Suppose there are N balls labeled 1, 2, 3, ... , N, and N boxes also labeled 1, 2, 3, ..., N.
If you put one ball inside of each box, how many combinations are possible so that there are no match of the ball label and box label (ball 1 is not in box 1, ball 2 is not in box 2, and so on)?
Do a web search for derangements.
Derangement -- from Wolfram MathWorld
If $N>3$, then you will find that the answer is approxminately $\frac{N!}{e}$.

4. Originally Posted by Soroban
Hello, lcmarincek!

Is this a homework problem?

If so, several questions arise:

. . First of all, who assigned it?
. . What course are you in?
. . What is your mathematical background?

. . How "good" is your teacher?
. . Would he/she recognize the right answer if shown it?

This is a complex problem with a highly elusive solution.
I'd prefer not to explain it ... if it goes pffft! over your heads.

Such an arrangement is called a "derangement of $N$ objects."

I'll let someone else explain it.
This isn´t actually a homework problem. It´s part of a problem I was trying to solve (just for fun).
At first glance it didn´t seem so complex. I started working on it, but I couldn´t find a trivial solution (neither a complex one...), but I still thought it had a not so advanced solution. That´s why I posted it on high school section.
But you mentioned the key to this problem: derangement. As far as I could see, it´s not a high school level problem.
I´ll focus my search on derrangement.
Thanks for the clue.