2005 AMC 12A Problem 25

Let S be the set of all points with coordinates (x,y,z), where x, y, and z are each chosen from the set \{0, 1, 2\}. How many equilateral triangles have all their vertices in S?

\textbf{(A) } 72 \qquad \textbf{(B) } 76 \qquad \textbf{(C) } 80 \qquad \textbf{(D) } 84 \qquad \textbf{(E) } 88

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