Maximum of a product of binomial coefficients

binomial-coefficientscombinatoricssequences-and-series

Let $x,b,\ell$ non negative integers, with $\ell\le b.$ Consider $b,\ell$ fixed. Let also $$F_b(x) = \binom{b}{x}\binom{b}{\ell-x}.$$ The maximum of $F_b(x)$ for $x=0,1,2,\dots, \ell$ is $F_b(\lfloor\ell/2\rfloor).$ A proof of this fact can be done in two steps.If
$$G_b(x)=\frac{F_b(x)}{F_b(x+1)}$$ for
$0\leq x\leq \lfloor \ell/2\rfloor-1$ then we can prove that $G_b(x)$ is increasing so $G_b(x)\le G_b(\lfloor \ell/2\rfloor-1).$ Also we can prove that $G_b(\lfloor \ell/2\rfloor-1)\leq 1.$ Now follows that $\max F_b(x) =F_b(\lfloor \ell/2\rfloor).$ The previous are elementary but a bit technical.
Although, I believe there is a simpler proof and not so technical. Any idea?

Best Answer

Your function is related to the probability mass function of the hypergeometric distribution provided that you set in that page:

  • $N=2b$ the population size,
  • $K=b$ the number of success states in the population,
  • $n=l$ the number of draws (i.e. quantity drawn in each trial),
  • $k=x$ the number of observed successes.

Under the section "Combinatorial identities" of the cited page you find the following identity:

$$\frac{{K \choose k}{N-k \choose n-k}}{{N \choose n}}=\frac{{n \choose k}{N-n \choose K-k}}{{N \choose K}}$$

which in your case is:

$$\frac{{b \choose x}{2b-b \choose l-x}}{{2b \choose l}}=\frac{{l \choose x}{2b-l \choose b-x}}{{2b \choose b}}$$

Now we know that both factors of the numerator of the RHS are maximized with $x = \lfloor l/2 \rfloor$ or $x = \lceil l/2 \rceil$.

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