I need to count how many handshakes happen between politicians and between politicians and lawyers. I can use the formula for the number of handshakes in a group of n people: n(n-1)/2. This formula works because each person shakes hands with n-1 others, and I divide by 2 to avoid double counting. For the politicians, n = 4, so they shake hands 4(4-1)/2 = 6 times. For the lawyers, each one shakes hands with 4 politicians, so there are 4 x 3 = 12 handshakes between lawyers and politicians. Adding these two numbers, I get 6 + 12 = 18 handshakes in total. 
Four politicians and three lawyers attend a party. Each politician shakes hands exactly once with everyone, and each lawyer shakes hands exactly once with each politician. How many handshakes take place?