[14 points] Suppose an integer $n > 1$ is such that $n + 1$ is not a multiple of 4 (i.e., such that $n$ is not congruent to $3 \bmod 4$). Prove that there exist $1 \leq i < j \leq n$ such that the following is a perfect square.
$$\frac { 1 ! 2 ! \cdots n ! } { i ! j ! }$$
Hint (use this or your own method): Make cases and first treat the case $n = 4k$.