If I have $8$ red balls and $8$ blue balls, what are the total number of arrangements possible where each row and each column have $2$ red and $2$ blue balls in a $4 \times 4$ grid?

How will this answer differ if all the ball were distinct?

For the first part, an idea was to find the solutions to $\sum\_{i = 1}^4 x\_{ij}= 0$, $j = 1, \cdots, 4$, $\sum\_{j = 1}^4 x\_{ij} = 0$, $i = 1, \cdots, 4$ where $x\_{\*\*} = \pm 1$, but that doesn't seem simple to solve.