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 objects."Suppose there are balls labeled 1, 2, 3, ... , N
and 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?
I'll let someone else explain it.