so this is a variant problem of the airplane seating. it states, suppose there are 100 people on an airplane that sit randomly in a seat, and ignore their own assigned seat. what is the probability at least one sits in their assigned seat.
I am setting up the inclusion/exclusion rule, but am having a bit of trouble setting this up. so far
let $A_i$ be the event where person A sits in their assigned seat. For a test case, let's examine the case where at most 1 person sits in assigned seat. For N=4, one person sits in their assigned seats, leaving 3 people to make mistakes. (1)(2)(2)(1) is the total choices over 4!. leaving $\frac{1}{(n-1)!}$. ?
Thoughts
I will work on the complement set instead.
First count all the possible seating arrangment, then subtract away the case where no one sits in their assigned seat.
The number of cases where no one sit at their own assigned seat is known as the derangement, $$D_n = \lfloor \frac{n!}{e}+\frac12\rfloor$$