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I know we can find the non negative integral solutions of the equation $x+y+z=24$ using Bars and Stars method. The same can be extended to provide the solutions for equations like $2x+y+z = 24$.

But is there any way to find the non negative integral solutions of

$3x + 2y + z = 24$

and subsequently a generalised formula of non negative integral solutions of

$nx + py + qz = k$ where $n,p,q>1$?

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I don't have an exact solution, but since the $d$-simplex $\sum_{i=1}^da_ix_i\le k$ has volume $k^d/(d!\prod_ia_i)$ for $a_i,\,k\ge0$, $\sum_{i=1}^da_ix_i=k$ should have $\sim k^{d-1}/((d-1)!\prod_ia_i)$ solutions.

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    $\begingroup$ It do gives 8.333 ~= 8 which is the right answer. But do you have any links of the proof ? $\endgroup$
    – Jdeep
    Jul 28, 2020 at 14:41
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    $\begingroup$ Okay, This is formula amazingly works!!!. I now need the proof $\endgroup$
    – Jdeep
    Jul 28, 2020 at 14:46
  • $\begingroup$ @NoahJ.Standerson For large $a_i,\,k,\,d$, it probably won't round to the exact value. $\endgroup$
    – J.G.
    Jul 28, 2020 at 14:48
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    $\begingroup$ @NoahJ.Standerson The fractional error will be $O(1/k)$, but it's hard to be more detailed than that. $\endgroup$
    – J.G.
    Jul 28, 2020 at 15:28
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    $\begingroup$ @NoahJ.Standerson The proof is on mathworld.wolfram.com/MultinomialDistribution.html $\endgroup$ Aug 1, 2020 at 14:04

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