Is there a formula for the coefficient of $\prod_{i=1}^{n}a_{i}^{n_i}\prod_{j=1}^{m}b_{i}^{m_i}$ in the product $$\prod_{i=1}^{n}\prod_{j=1}^{m}\left(a_{i}+b_{j}\right).$$ Is there a simple way to bound these coefficients?
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$\begingroup$ The general product is somewhat symmetric, and the monomials will have the property that each of them has "length" mn. (Also, each exponent is constrained.) After you handle those restrictions, you then use a multinomial coefficient which counts how often you pick each symbol from the appropriate set. Consider looking at multinomial coefficients for more general products. Gerhard "Higher Rank Than Corporal Product" Paseman, 2018.12.12. $\endgroup$ – Gerhard Paseman Dec 12 '18 at 17:49

1$\begingroup$ This reduces to the following combinatorial problem: given an $n \times m$ grid, how many ways are there to label the squares black or white, given the data of how many black/white squares are in each column and row? $\endgroup$ – Kevin Casto Dec 12 '18 at 17:59

$\begingroup$ @GerhardPaseman Thank you for your answer. I'm not sure I understood. Do you reckon the relevant coefficient is simply the multinomial the choose n_i 1<=i<=n and m_j 1<=j<=m from nm? I looked for your suggested topic and did not find relevant results, can you share a link, please? $\endgroup$ – MathGirl88 Dec 13 '18 at 7:36

$\begingroup$ @KevinCasto , Yes, I believe its the same. $\endgroup$ – MathGirl88 Dec 13 '18 at 7:37