Combinatorics - sending letters

There were $200$ letters sent in total. So at least one person must have received at least $10$ letters (because if everyone got at most $9$ letters, then only at most $180$ letters were received).

Say that Bob received at least $10$ letters. Since there are $19$ people who are not Bob, there are at most $9$ other people who didn't send a letter to Bob. Since Bob sent $10$ letters, he must have sent a letter to one of the people who sent one to him.


Draw a directed graph where an arrow from a to b means a sent letter to b. The average in degree is the average out degree and this is 10. So at least one person P get letters from 10 different people and so one of the persons sending a letter to P also get a letter from P.

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