Question

Everybody in a room shakes hands with everybody else. The total number of hand shakes is $66$. The total number of persons in the room

Answer 1

For a handshake 2 people are required who will shake hand with each other

Suppose there are n people in the room

So we need to find the number of ways to make a pair out of n persons

which is

${}^n{C}_2=66\Rightarrow \frac{n!}{(n-2)!2!}=\frac{n(n-1)}{2}$

$\frac{n(n-1)}{2}=66\Rightarrow n^2-n-132=0\Rightarrow n^2-12n+11n-132=0\Rightarrow n(n-12)+11(n-12)=0\Rightarrow (n+11)(n-12)=0\Rightarrow n=12,-11$

Correct Answer is 12 as $n$ cannot be negative