Every body in a room shakes hands with every body else. The total number of hand shakes is 66. The total number of persons in the room is

Let total number of persons be n. Since two persons shake hands, so number of ways of shaking hands ${=}^{\mathrm{n}}{\mathrm{C}}_2$.

According to question, we have ${ }^{\mathrm{n}}{\mathrm{C}}_2=66$

or $\frac{\mathrm{n}!}{(\mathrm{n}-2)!2!}=66\text{ or }\frac{\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)!}{(\mathrm{n}-2)!2\times 1}=66$

or ${\mathrm{n}}^2-\mathrm{n}=132\text{ or }{\mathrm{n}}^2-\mathrm{n}-132=0$

or ${\mathrm{n}}^2-12\mathrm{n}+11\mathrm{n}-132=0$

or $\mathrm{n}(\mathrm{n}-12)+11(\mathrm{n}-12)=0\Rightarrow \mathrm{n}=-11\text{ or }12$

But, n = –11 is not possible.

$\therefore \mathrm{n}=12\text{. }$