A probability problem by Abhilash Yadav

Let the number of people in the party be $n$ , then the number of handshakes is given by combinations of $2$ out of $n$ which is given by: $\left( \begin{matrix} n \\ 2 \end{matrix} \right)=\frac{n(n-1)}{2}$

Since the total of handshakes is $66$ , therefore: $\Rightarrow \frac{n(n-1)}{2}=66\Rightarrow n = \boxed{12}$

The problem is quite simple.......lets say we have four people....then the first one shakes hands with 3 people....the second with 2.....the third with one.....Thus Extending the same analogy we must have nC2 choices for the no.of choices in the n people.......by using this formula....and solving the quadratic we have 12,-11....as no.of people can't be negative...so Ans. is 12 people.