Chance n uniforms add up to at most one

Let $X_1,\dots,X_n$ be independent, each uniform on $(0,1)$. Find $\mathbb{P}(X_1+X_2+\dots+X_n\le1)$.

$1/n!$. The event is the simplex $\{x_i\ge0,\ \sum x_i\le1\}$ inside the unit cube, whose volume is $\tfrac1{n!}$. By induction, $\mathbb{P}(S_{n-1}\le s)=\tfrac{s^{n-1}}{(n-1)!}$ for $s\le1$, and conditioning on $X_n=x$ gives $\int_0^1\tfrac{(1-x)^{n-1}}{(n-1)!}dx=\tfrac1{n!}$.