Some similar puzleson inclusion-exclusion

Inclusion exclusion is a common technique for lost of combinatorics puzzles. Here are two I recently encountered:

0. At the banquet of a large conference, n mathematicians hang their coats on the coat rack as they enter. At the end of the night they leave in a drunken stupor, each one randomly putting on a coat without checking that it’s their own. Show that in the limit as n → ∞, the probability that none of the mathematicians staggers home in their own coat approaches 1/e. 

1. 6 persons are standing in a line what is the probability that no three consecutive people are in increasing order of their heightsalso see: https://artofproblemsolving.com/wiki/index.php/Principle_of_Inclusion-Exclusion

2. 6 babies are born in a hospital on either monday, tue, wed, thu. What is the probability that there was no day when no baby was born?

3. 10 people numbered uniquely in 1-10 are uniformly randomly given 10 tickets uniquely numbered b/w [1,10]. What is the probability that none of them get the ticket with same number as the number assigned to them.