2010 AMC 12B Problem 15

For how many ordered triples (x,y,z) of nonnegative integers less than 20 are there exactly two distinct elements in the set \{i^x, (1+i)^y, z\}, where i=\sqrt{-1}?

\textbf{(A)}\ 149 \qquad \textbf{(B)}\ 205 \qquad \textbf{(C)}\ 215 \qquad \textbf{(D)}\ 225 \qquad \textbf{(E)}\ 235

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.