at a United Nations meeting, a group of 15 delegates meet to discuss an international peace treaty. If all of the delegates shake hands with one another, how many handshakes are exchanged?

That should be straight forward. In a group of n people each person shakes hands with the n-1 other people. That would be a total or n(n-1) except that each had shake involves two people. The total number of hand shakes is n(n-1)/2.