Doubt--------------

Let no. of Total persons at party = x

This is a porblem on combination bcoz each person will shake hand with other one. Suppose two of them are A & B. Here if we apply permutations then there will be included a case in which A will shake hand with B & B will shake hand with A, which is not possible & irrelevant. So, here we will apply combination as follows- 

xC2 = 66

this implies (x(x-1))/2= 66

implies x= 12