Hi there,(adsbygoogle = window.adsbygoogle || []).push({});

I know that the number of solutions of [itex]a_{1}+a_{2}+...a_{n}=k [/itex] where [itex]a_{i} [/itex] are non-negative integers is just [itex]\binom{n+k-1}{k} [/itex].

Now, given a [itex]m\times n [/itex] matrix with non negative integers [itex]a_{ij}[/itex], how many solutions are there for the following kind of system?

[itex]\begin{array}{ccccccccc}

s_{1} & & s_{2} & & & & s_{n}\\

\shortparallel & & \shortparallel & & & & \shortparallel\\

a_{11} & + & a_{12} & + & ... & + & a_{1n} & = & k_{1}\\

+ & & + & & & & +\\

a_{21} & + & a_{22} & + & ... & + & a_{2n} & = & k_{2}\\

+ & & + & & & & +\\

\vdots & & \vdots & & & & \vdots\\

+ & & + & & & & +\\

a_{m1} & + & a_{m2} & + & ... & + & a_{mn} & = & k_{m}

\end{array} [/itex]

The sum of each row and each column is constrained by the known non-negative integers [itex]k_{i}[/itex] and [itex]s_{i}[/itex].

Without the column restriction we would have [itex]\prod_{i=1}^{m}\binom{n+k_{i}-1}{k_{i}} [/itex].

Thanks a lot.

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# Number of solutions

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