I was trying to prove the following statement using probabilistic methods: Given that $G$ is a graph on $n\geq 10$ vertices, is a graph that has the property: If we draw a new edge, then the number of copies of $K_{10}$ increases. Prove that $|E|\geq 8n-36$.
I am interested in learning the subject but I am stuck in this problem. The idea that I have is that I should perhaps find some event $X$ such that $P(X)\geq (8n-36)/||E|$, but I dont know what should the event be.
Hints on how to start this problem would be greatly appreciated.