Combinatorics problems that can be solved more easily using probability 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.  

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
