Hi Nazrul,

Suppose there were $n$ people at the party including you and me. I shook hands with $n-1$ people, so did you and so did everyone else at the party. Thus it looks like there were $n(n - 1)$ handshakes but this included you shaking hands with me and me shaking hands with you. In fact it counts each handshake twice so there are actually $\large \frac{n(n - 1)}{2}$ handshakes.

Can you complete the problem now? Can you do it without solving a quadratic?

Penny