Suppose that each of $n$ men at a party throws his hat into the center of the room. The hats are first mixed up, and then each man randomly selects a hat. What is the probability that
(a) None of the men selects his own hat;
In this question, I know that the answer given uses Principle of Inclusion-Exclusion (PIE), but I have the following solution. Could someone please explain why this does not work and how I am supposed to figure out for other such questions when to use my way, and when to use PIE?
|Probability Space| = n! ways of arranging |Event that every person gets any hat that is not theirs| = Step 1: Person 1 picks any hat other than theirs in n-1 ways Step 2: Person 2 picks any hat other than theirs and the hat already picked in (n-1)-1=n-2 ways...and so on. Total cardinality for the event = (n-1)!
Therefore P = (n-1)!/n!
What if the first person picked the second person's hat and then the second person has got $n-1$ instead of $n-2$ ways to pick a hat that is not his own.