I was hoping to get some help in understanding the solution provided below. I understand that the integral will evaluate to $\frac{1}{n!}$, however, my uncertainty lies in how the integral is setup. I am not sure how we use $C$ to setup the integrands. Why would it not be $${\int_0^1 \int_0^{x_n} \dotsb \int_0^{x_3} \int_0^{x_2} \int_0^{x_1}} ds_1 ds_2 \dotsc ds_n?$$. Let $C$ be a set in $n$-dimensional space and let
$$Q(C) = \underset{C}{\int\!\!\dotsb\!\!\int} dx_1 dx_2 \dotsc dx_n.$$
If $C = \{(x_1, x_2, \dotsc, x_n) : 0 \le x_1 \le x_2 \le \dotsb \le x_n \le 1 \}$, then
$$Q(C) = {\int_0^1 \int_0^{x_n} \dotsb \int_0^{x_3} \int_0^{x_2}} dx_1 dx_2 \dotsc dx_n = \frac{1}{n!}.$$