At a party , each person shakes hands with every other person . the total number of hand shakes was 66. how many persons are present at the party ?
A 6 b 11 c 12 d 18 e 33
Line up half of the people with the other half of the people facing them.
One line has $\displaystyle \frac{x}{2}$ people, so $\displaystyle \frac{x}{2}$ handshakes can take place.
Each person can shake hands with (x-1) other people,
hence there are $\displaystyle (x-1)\frac{x}{2}$ handshakes possible
$\displaystyle (x-1)\frac{x}{2}=66$
$\displaystyle (x-1)x=132$
the factors of 132 that differ by 1 are 11 and 12, hence x=12, x-1=11.
12 people are at the party.