TITLE: Combinatorics problems that can be solved more easily using probability
QUESTION [5 upvotes]: I'm looking for examples of combinatorics problems which would be very difficult to solve by direct enumeration, but can be easily solved using ideas from probability, like independence, commuting sums and expectations, etc.  I know I have seen such problems before, especially in the HMMT combinatorics subject tests, but I can't now recall any good examples.  
I am NOT looking for probabilistic existence proofs (the so-called "probabilistic method" introduced by Erdos).  The sort of problems I'm interested in are enumerative.

REPLY [3 votes]: Here are some problems:

Find the sum of the number of all continuous runs of all possible sequences with $2019$ ones and $2019$ zeros
https://artofproblemsolving.com/community/c6h366278p2018435
https://artofproblemsolving.com/community/c6h60752p366512
https://artofproblemsolving.com/community/q2h1151650p5452212
http://artofproblemsolving.com/community/c6h1170845p5622460
https://artofproblemsolving.com/community/q4h1497881p9354029
https://artofproblemsolving.com/community/q1h79679p10589332