In there are $12$ persons in a party, and if each two of them shake hands with each other, how many handshakes happen in the party ?

The correct option is C. $66$
The total number of handshakes is same as the number of ways of selecting $2$ persons among $12$ persons as out of $12$, each $2$ of them are shaking hand.

$\therefore$ No. of possible ways $={}^{12}{C}_2=\frac{12!}{10!\times 2!}=66$ [${}^n{C}_r=\frac{n!}{r!(n-r)!}$]