2012 AMC 12B Problem 25

Let S=\{(x,y) : x\in \{0,1,2,3,4\}, y\in \{0,1,2,3,4,5\},\text{ and } (x,y)\ne (0,0)\}. Let T be the set of all right triangles whose vertices are in S. For every right triangle t=\triangle{ABC} with vertices A, B, and C in counter-clockwise order and right angle at A, let f(t)=\tan(\angle{CBA}). What is \prod_{t\in T} f(t)?

\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{625}{144}\qquad\textbf{(C)}\ \frac{125}{24}\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ \frac{625}{24}

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