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How to count the number of integer solutions to $$\sum_{i=i}^{n}{f_ig_i} \geq 5$$ such that $\displaystyle \sum_{i=1}^{n}{f_i}=6$ , $\displaystyle \sum_{i=1}^{n}{g_i}=5$ , $\displaystyle 0 \leq f_i \leq 4$ , and $\displaystyle 0 \leq g_i \leq 4$?

Is there a general formula to calculate things like this?

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    Is $n$ fixed?...2012-05-17
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    Yes, It is. Assume n=20.2012-05-17
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    Looks like a hard question. Why don't you start with something smaller (that is, replace all the numbers with numbers small enough that you can easily write out all the solutions) and see whether there are any patterns you can exploit?2012-05-18
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    @Gerry, not for a computer!2012-05-18
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    @Yuval, provided the computer has someone like you around to program it!2012-05-18

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