P(k=n)=\int_0^1dt\int_0^tdx_1\int_0^{t-x_1}dx_2\cdots\int_0^{t-(x_1+\cdots+x_{n-2})}t\cdot dx_{n-1}=\int_0^1tdt\int_0^tP_{n-2}(t-x_1)dx_1=\int_0^1tdt\int_0^tP_{n-2}(x_1)dx_1=\frac{1}{n(n-2)!}.