Asymptotic behavior of a sum

Asymptotic behavior of a sum

This is the extension of my previous question. Let $A= \sum_{x_1=0}^m \sum
_{x_2=0}^m \cdots \sum_{x_n=0}^m \min_r(x_1, x_2,.., x_n)$ where $\min_r$
is the $r^{th}$ minimum of $(x_1, x_2,.., x_n)$. For example if $x_1\leq
x_2 \leq \cdots x_r \leq \cdots $ then $\min_r(x_1, \ldots, x_n)=x_r.$
Now let $B =\sum_{k=0}^m n \binom{n-1}{r-1}k^{r-1}(m-k)^{n-r}k.$ Is it
true $\frac{A}{B} \rightarrow 1$ as $m $ approaches to infinity?