TITLE: How many handshakes in total?
QUESTION [5 upvotes]: If there are 20 persons in a party and if each two of them shake hands with each other, 
how many hand shakes happen in the party ? 
will it be 20C2 = 20*19 or 19+18+...+1? there are 2 different explanation and both seem to be logically correct
i said it 20C2 because suppose there re 4 people. A B C D. so the possible hand shakes are
{(AB), (AC), (AD), (BC), (BD), (CD)} =6 = 4c2
another way is 19+18+..+1. can somebody explain? answer is 380 or 190?

REPLY [2 votes]: Every person shakes hands with $19$ persons, so at first sight there are $20\times19=380$ handshakes. But by every handshake two persons are involved. So $380$ is the result of double-counting. There are $190$ handshakes.