If every person at a party shakes the hand of every other person and there were 105 handshakes in all How many persons were present at the party?

The answer is 15 people. Each shook hands with 14 others, and there are half that many handshakes (pairs).

The total number of pairs (distinct handshakes) within the group is defined by the formula T = [n!/(n-2)!] /2

Given T = 105

we get n!/(n-2)!=210 which implies n(n-1)=210 on solving we get n=15