What is the volume of the region in $R^n$ defined as follows
$$ V_n = \{(x_1,x_2, \dots, x_n) \in R^n | x_i \ge 0 \text{ and } \sum x_i \leq 1 \}$$
Scroll down for a clever proof (if you know the source, please comment)
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Transform the space with $y_j = \sum_{1 \le i \le j} x_i$
This is a linear transformation that gives a new region defined as
$$ W_n = \{(y_1,y_2, \dots, y_n) \in R^n | 0 \le y_1 \le y_2 \le \dots \le y_n \le 1 \}$$
$W_n$ is just a subset of the hypercube $[0,1]^n$. The hypercube can be split into $n!$ regions of equal volume, each region corresponding to a sort order among the coordinates. $W_n$ is one of them.
Thus volume of $W_n$ is $\frac{1}{n!}$
Since the determinant of the linear transformation from $V_n$ to $W_n$ is $1$, the volume of $V_n$ is $\frac{1}{n!}$ too!
