I want to verify the result that I found to this equation:
$\prod_{i=1}^{n}\prod_{j=1}^{2} ij$
I found that $\prod_{j=1}^{2} ij = 2i^2$. After that I did: $\prod_{i=1}^{n} 2i^2$ and, finally, found that:
$\prod_{i=1}^{n}\prod_{j=1}^{2} ij = 2^n (n!)^2$
I am not sure if I can solve it this way (a product first and then another product).
This is exactly what is expected. You can imagine parentheses around the inner product. You should evaluate that first, as you did.