A hall has n doors. Suppose that n people each choose any door at random to enter the hall.

What is the probability that at least one door will not be chosen by any of the people.

Here is my attempt:

all possible combination: n^n

P(at least one door not chosen) = 1- P(all doors chosen)

= 1 - n!/n^n

However, this is not what solution has, rather it contains

(n-1)^n + (n-2)^n + ... + 1^n / n^n

Wheres my flaw?