Problem to find arithmetic mean of the largest elements of $r$-subsets of ${1,2,...,n}$

Observe that

$$(r+k)\binom{r+k-1}{r-1}=\frac{(r+k)!}{(k!)(r-1)!}\cdot \color{blue}{\frac{r}{r}}=\underbrace{r\binom{r+k}{k}=r\binom{r+k}{r}}_{\binom{a}{b}=\binom{a}{a-b}}.$$ Thus $$r\binom{r-1}{r-1}+(r+1)\binom{r}{r-1}+\dotsb+n\binom{n-1}{r-1}=r\sum_{k=0}^{n-r}\binom{r+k}{r}$$

Now use the Hockey-stick identity to get $$r\sum_{k=0}^{n-r}\binom{r+k}{r}=\color{red}{r\binom{n+1}{r+1}}.$$ So

$$H(n,r)=\frac{r\binom{n+1}{r+1}}{\binom{n}{r}}=r\left(\frac{n+1}{r+1}\right).$$