Instead, I will try to use some combinatorics to simplify the counting. I know that the number of ways to choose two items out of n items is given by the formula n choose 2, which is equal to n(n-1)/2. So, to count the number of handshakes among the gremlins, I can use 20 choose 2, which is 20(19)/2 = 190. The imps did not shake hands with each other, so there are no handshakes to count among them. Therefore, the total number of handshakes at the convention is 190 + 300 = 490. 
Twenty gremlins and fifteen imps are at the Annual Mischief Convention. The imps have had a lot of in-fighting lately and refuse to shake hands with each other, but they readily shake hands with all of the gremlins. Meanwhile, all the gremlins are quite friendly and shake hands with all of the other gremlins as well as imps. Each pair of creatures shakes hands at most once. How many handshakes were at the convention?