2003 AMC 12A Problem 23

How many perfect squares are divisors of the product 1! \cdot 2! \cdot 3! \cdot \cdots \cdot 9!?

\textbf{(A)}\ 504\qquad\textbf{(B)}\ 672\qquad\textbf{(C)}\ 864\qquad\textbf{(D)}\ 936\qquad\textbf{(E)}\ 1008

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