Consider $$ \Omega_N:=\left\{0\leq x_1\leq x_2\leq\ldots\leq x_N\leq L;~0\leq y_1\leq y_2\leq\ldots\leq y_N\leq L\right\}. $$
It is said that the volume of $\Omega_N$ is $$ V_N=V(\Omega_N)=L^{2N}/(N!)^2. $$
How to see this?
Moreover, it is said that any integration over $\Omega_N$ may be replaced by an integration of the symmetrized integrand over $L^{2N}$ and multiplication of the result by $(N!)^{-2}$,
$$ \int_{\Omega_N}f(C_N)dx^Ndy^N=\frac{1}{(N!)^2}\int_{L^{2N}}f_{\text{sym}}(C_N)dx^N dy^N. $$ where $C_N$ are configurations.
I do not understand that. Do you have any idea how to get this? I literally have no idea.