Please use pigeonhole principle to solve this problem d Ther

Please use pigeonhole principle to solve this problem.

d) There are 10 people in the class. At the end of the midterm, each sent an email to exactly 5 of them. Prove that there are two people who sent emails to each other. Please use pigeonhole principle to solve this problem.

Solution

There are 10 people in the class.
So there are ¹⁰C₂ exclusive pairs of two i.e. max. 45 no. of emails can be sent without two people sending emails to each other.
But the emails being sent are 10*5= 50
So, there are 50 pigeons and 45 pigeonholes.
That implies there must be at least one pair who sent emails to each other.
Hence proved.