Everybody in a room shakes hands with everybody else.

Let us suppose there are n persons in a room for a handshake, 2 people are requried who will shake hand with each other.

So, number of ways to make a pair out of n persons is nC2 = 66 (given) 

i. $\mathrm{e} \frac{n !}{2 !(n-2) !}=66$

i. $\mathrm{e} \frac{n(n-1)(n-2) !}{2 \times(n-2) !}=66$

i.e $n^{2}-n=132$

i.e $n^{2}-n-132=0$

i. e $n^{2}-12 n+11 n-132=0$

i.e $n(n-12)+11(n-12)=0$

i.e $(n-12)(n+11)=0$

i.e $n=12$ or $n=-11$

Since n = −11 is not possible  

⇒  n = 12

Hence, the correct answer is option B.