An interesting combinatorial identity is the Vandermonde convolution identity:


$$ \sum\_k {n\choose k}{m\choose s-k} = {n+m \choose s},$$


which can be proved by considering the coefficients in $(x+1)^{n+m} = (x+1)^n (x+1)^m$. I am interested in the behavior of the summands: which summands among them contribute the most?.


More specifically, I'd like to look at the behavior of the function


$$ F(k) = \frac{{n\choose k}{m\choose s-k}}{{n+m \choose s}}. $$


In a specific probabilistic problem, intuition suggests that $F(k)$ peaks near $k$, with $\frac{n}{n+m} = \frac{k}{s}$. Indeed, in the limit case $s = n+m$, $k$ had better be $n$ for the value to survive. However, I'm not satisfied with the limit case.


Given that $F$ does peak around $k = k^\* := \frac{ns}{n+m}$, I want to know how much it peaks. Thus the following **question**:



> 
> Find the smallest possible non-negative integer $\epsilon$ such that
> 
> 
> $$ \sum\_{k \in [k^\*-\epsilon, k^\*+\epsilon]} F(k) > 90\%.$$
> 
> 
> 


The answer $\epsilon$ depends on $n, m, s$, and "$90\%$".



Local behavior of the Vandermonde convolution
$\newcommand\ep{\epsilon}$


Of course, there is no simple explicit expression for the smallest $\ep$. However, one can give upper bounds on or approximations of this $\ep$. 
Indeed, $F$ is the probability mass function of a random variable, say $X$, with the [hypergeometric distribution](https://en.wikipedia.org/wiki/Hypergeometric_distribution) with parameters $n,m,s$. So, by a [Hoeffding inequality](https://en.wikipedia.org/wiki/Hypergeometric_distribution#Tail_bounds), for $\ep\in[0,k^\*]$
$$S(\ep):=\sum\_{k\in[k^\*-\ep,k^\*+\ep]}F(k)=P(|X-k^\*|\le\ep)
\ge1-2e^{-2\ep^2/s}.$$
Solving now the equation $1-2e^{-2\ep^2/s}=0.90$ for $\ep>0$, we get a desired upper bound on the smallest $\ep$ such that $S(\ep)\ge0.90$. 


If $s$ is much smaller than $\min(n,m)$ and if $\frac n{n+m}$ is not close to $0$ or $1$, then one can use a [normal approximation to hypergeometric distribution](https://en.wikipedia.org/wiki/Hypergeometric_distribution#Related_distributions).


