Sum of homogeneous products

Prove that the sum of the homogeneous products of $r$ dimensions of the numbers $1,2,3,\dots n$, and their powers is $\frac{(-1{)}^{n-1}}{(n-1)!}\left\{1^{n+r-1}-\frac{n-1}{1}{2}^{n+r-1}+\frac{(n-1)(n-2)}{1\cdot 2}{3}^{n+r-1}-\dots \text{to }n\text{ terms}\right\}$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Miscellaneous Examples, No. 295