SOLUTION: There was a party and everyone shook everyone's hands. There were 66 handshakes!! How many people were at the party???????????

Question 108982: There was a party and everyone shook everyone's hands. There were 66 handshakes!! How many people were at the party???????????

Answer by MathLover1(20849) (Show Source): You can put this solution on YOUR website!
There was a party and everyone shook everyone's hands. There were 66 handshakes!! How many people were at the party?

we know that:
with two people (x and y), there is handshake; ( with )
with three people (x,y, and z), there are handshakes; x with y and z, and y with z

with four people (x, y, z, and e), there are handshakes; x with y, z, and e, then y with z and e, then z with e
In general:
with people, the number of handshakes of the
first consecutive numbers:
Since this is ,
we need to solve the equation ...multiply both sides by .=> ....=>...
this is the quadratic equation
use quadratic formula and solve for :

since }, , and , you will have:

we need only positive root:

we obtain as the answer and deduce that .

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