How many $4 \times 4$ arrays whose entries are $0$s and $1$s are there such that the row sums (the sum of the entries in each row) are $1, 2, 3,$ and $4,$ in some order, and the column sums (the sum of the entries in each column) are also $1, 2, 3,$ and $4,$ in some order? For example, the array
\[\left[   \begin{array}{cccc}     1 & 1 & 1 & 0 \\     0 & 1 & 1 & 0 \\     1 & 1 & 1 & 1 \\     0 & 1 & 0 & 0 \\   \end{array} \right]\]
satisfies the condition.
$\textbf{(A) }144 \qquad \textbf{(B) }240 \qquad \textbf{(C) }336 \qquad \textbf{(D) }576 \qquad \textbf{(E) }624$
Note that the arrays and the sum configurations have one-to-one correspondence. Furthermore, the row sum configuration and the column sum configuration are independent of each other. Therefore, the answer is $(4!)^2=\boxed{\textbf{(D) }576}.$
~bad_at_mathcounts ~MRENTHUSIASM