A problem by Sujeet Srivastava

There can be 2 solutions:

Method 1: let there be n people

Choose 2 out of n and have a handshake

nC2=66

n(n-1)/2=66

n=12

Method 2: first person shakes hand with (n-1) others

2nd person shakes hand with (n-2) people [as he has already with person 1]

And so on.. So (n-1)+(n-2)+.....1

=(n-1)(n-1+1)/2=66

Again same eqn, n=12

In general, with n+1 people, the number of handshakes is the sum of the first n consecutive numbers: 1+2+3+ ... + n. Since this sum is n(n+1)/2, we need to solve the equation n(n+1)/2 = 66. This is the quadratic equation n2+ n -132 = 0. Solving for n, we obtain 11 as the answer and deduce that there were 12 people at the party.