n students have n identical raincoats which they unwittingly hang on the same coat rack while attending the class. After class, each student selects a raincoat at random, being unable to tell it apart from all the others. Find the expected number of raincoats that ends up with its original owner.
There is chance n! arangements of n coats between n students. Therefore there chance that nth student will end up with his coat is equal to 1/n!, for (n-1)th student chance is equal to 1/(n-1)! and so on. Therefore expected number of raincoats is equal to n*1/n! + (n-1)*1/(n-1)! + ... + 1.