# A curious inequality concerning binomial coefficients

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 $$A = \sum_i a_i$$. 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 $$B$$.

• Take $a= (30,15,3), b = (20,9,2)$ to see that the inequality does NOT hold. – Dr. Wolfgang Hintze Feb 16 at 8:56
• Gah. Sorry about that. I have edited the question. – Navin K. Feb 16 at 9:06