At a party, everyone shook hands with everybody else. There were 66 handshakes. How many people were at the party?

To see this, imagine that  just 4 people were at the party

The first person shakes 3 hands

The second person shakes 2 hands

The third person shakes 1 hand

The last person has already shaken hands with everybody

So...the number of handshakes between 4 people =  (4)(3) / 2 =  6

So....it appears that the number of handshakes between n people  is just n(n - 1) /2

So.....we can solve this

n(n - 1) / 2 = 66      multiply both sides by 2

n(n - 1) = 132   simplify

n^2 - n = 132

n^2 - n - 132 = 0     factor

(n - 12)  ( n + 11) = 0

Setting each factor to 0  ans solving for n  ives n = -11  (reject)   or n = 12 (accept)

So....there wqere 12 people