2022 AMC 12B Question 17

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 $$
\begin{bmatrix} 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \end{bmatrix}
$$ satisfies the condition.