2015 Cayley Problem 24

In the diagram, a rectangular ceiling PQRS measures 6 m by 4 m and is to be completely covered using 12 rectangular tiles, each measuring 1 m by 2 m. If there is a beam, TU, that is positioned so that PT = SU = 2 m and that cannot be crossed by any tile, then the number of possible arrangements of tiles is

\textbf{(A)}\ 180\quad \textbf{(B)}\ 190\quad \textbf{(C)}\ 185\quad \textbf{(D)}\ 170\quad \textbf{(E)}\ 175

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.