Assume we have the hyperplane $x_1+x_2+\ldots+x_N=S$ with $S\in R$ and $S>0$. Is there a formula giving/approximating the ratio between the area of the region satisfying the conditions: $0\leq x_i \leq M$ and that of the region satisfying $0\leq x_i \leq S$, where $S/N \leq M \leq S$?
I understand that we can see the region of the hyperplane $0\leq x_i \leq S$ as the simplex with N-1 dimensions, and that We can see the region I am looking for as as the intersection with a cube with a vertex in $O$ and edges $M$. I can solve the problem fixed $N$ ($N=2, 3$ is simple), but is there a general expression (even approximating) for this ratio for arbitrary $N$?