Can you shakehands with your friends

If there are 'n' people.

No. of handshakes = (n*(n-1))/2

Hence, n=12.

This is actually a combinatorics problem. I don't know what the HELL is it doing here in Number Theory. Anways, let us take the no. of people as $n$ . Now, we know that in a handshake, there are $2$ people and the arrangement of the people in a handshake doesn't matter, i.e., if the $2$ person's position are exchanged, then also the handshake is the same as the previous one and handshakes are not double counted like that. So, here we use combinations to find out the no. of people.

From all the above data, we can say that there are $nC2$ or $\binom{n}{2}$ handshakes as for a handshake, we select $2$ people out of $n$ people. Also, it is said that there had been $66$ handshakes. So --->

$\binom{n}{2}=66$

$\implies \frac{n!}{2!\times (n-2)!}=66$

$\implies \frac{n!}{(n-2)!}=66\times 2! = 66\times 2 = 132$

$\implies n(n-1)=132$

$\implies n^2-n-132=0$

$\implies (n-12)(n+11)=0$

$\implies n=12\quad or \quad n=(-11)$

But, no. of people cannot be (-ve), so $n\neq (-11)$ , so we finally have $n=\boxed{12}$

So, there are 12 persons in the room.

P.S. -- For those who dont know about combinations, I am giving a little tutorial here. Combinations is referred to the no. of ways in which $r$ things can be selected out of $n$ things at a time and that value is represented by $nCr$ or $\binom{n}{r}$ .

We have the formula as follows ---> $nCr=\binom{n}{r}=\frac{n!}{r!\times (n-r)!}$

$\frac{n!}{2!(n-2)!}=66$

$\frac{n!}{(n-2)!}=132$

$\frac{n(n-1)(n-2)!}{(n-2)!}=132$

$n(n-1)=132$

$n^{2}+n-132=0$

$(n-12)(n+11)=0$

$n=12$ or $n=-11$ , we choose the real number $\boxed{12}$

(n(n-1))/2 =66 , hence n=12

No. of handshakes = $\frac { n(n-1) }{ 2 }=66$ . . . So we get the quadratic equation of ${ n }^{ 2 }-n-132$ . . .which gives n = 12 and -11 . . . So, the answer is 12.

12C2=66

its just c(n,2) n*(n-1)/2!=66 n=12

let no. people be x then (x*x-1)/2 =12

let the no. of people be x then , No. of handshakes= (x*(x-1))/2) therefore x equals 12