In the middle of a proof in a graph theory book I am looking at appears the inequality $\sum_i {d_i \choose r} \ge n { m /n \choose r},$ and I'm not sure how to justify it. Here $d_i$ is the degree of vertex $i$ and the sum is over all $n$ vertices. There are $m$ edges. If it is helpful I think we can assume that $r$ is fixed and $n$ is large and $m \approx n^{2 - \epsilon}$ for some $\epsilon > 0$.
My only thought is that I can do it if I replace every ${d_i \choose r}$ by $(d_i)^r / r!$, but this seems a little sloppy.
Also, for my purposes, I am happy with even a quick-and-dirty proof that
$\sum_i {d_i \choose r} \ge C \, n { m /n \choose r}$ holds for some constant $C>0$.
Motivation: an apparently simple counting argument gives a lower bound on the Turán number of the complete bipartite graph $K_{r,s}$.
Source: Erdős on Graphs : His Legacy of Unsolved Problems. In the edition I have, this appears on p.36.
Relevant part of the text of the book:
3.3. Turán Numbers for Bipartite Graphs
Problem
What is the largest integer $m$ such that there is a graph $G$ on $n$ vertices and $m$ edges which does not contain $K_{r,r}$ as a graph? In other words, determine $t(n, K_{r,r})$.
.... for $2\le r \le s$ $t(n,K_{r,s}) < cs^{1/r}n^{2-1/r}+O(n) \qquad (3.2)$
The proof of (3.2) is by the following counting argument:
Suppose a graph $G$ has $m$ edges and has degree sequence $d_1,\ldots,d_n$. The number of copies of stars $S_r$ in $G$ is exactly $\sum_i \binom{d_i}r.$ Therefore, there is at least one copy of $K_{r,t}$ contained in $G$ if $\frac{\sum_i \binom{d_i}r}{\binom nr} \ge \frac{n\binom{m/n}r}{\binom nr}>s$ which holds if $m\ge cs^{1/r}n^{2-1/r}$ for some appropriate constant $c$ (which depends of $r$ but is independent on $n$). This proves (3.2)