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Has anyone seen an inequality of this form before? It seems to be true (based on extensive testing), but I am not able to prove it.

Let $a_1,a_2,\ldots,a_k$ be non-negative integers such that $\sum_i a_i = A$. Then, for any non-negative integer $B \le A$: $$ \sum_{(b_1,\ldots,b_k): \sum_i b_i = B} \prod_i \frac{\binom{a_i}{b_i}}{\binom{A-a_i}{B-b_i}} \ge {\binom{A}{B}}^{2-k}. $$ The sum on the left is over all tuples $(b_1,b_2,\ldots,b_k)$ of non-negative integers, with $b_i \le a_i$ for all $i$, whose sum is equal to $B$.

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By Cauchy–Bunyakovsky–Schwarz inequality we have $$ \left(\sum \prod_i \frac{\binom{a_i}{b_i}}{\binom{A-a_i}{B-b_i}}\right)\left(\sum\prod_i \binom{a_i}{b_i}\binom{A-a_i}{B-b_i}\right)\geqslant \left(\sum \prod_i\binom{a_i}{b_i}\right)^2=\binom{A}{B}^2 .$$

Thus it suffices to prove that $$ \sum\prod_i \binom{a_i}{b_i}\binom{A-a_i}{B-b_i}\leqslant \binom{A}B^k. $$ But RHS is just the sum of the same guys without the restriction $\sum b_i=B$.

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  • $\begingroup$ Brilliant! Thank you. $\endgroup$ – Navin K. Mar 13 '19 at 19:56

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