Question 10
Let $\mathcal{S}$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0 \le x \le 2,$ $0 \le y \le 3,$ and $0 \le z \le 4.$ Two distinct points are randomly chosen from $\mathcal{S}.$ The probability that the midpoint of the segment they determine also belongs to $\mathcal{S}$ is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$