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What is the number of integer solutions to the problem $$\sum_{i=1}^{i=k}x_i = n$$ subject to $\forall_i\ \ x_i \ge 0 $

note This should hold for both cases $k < n$ and $k \ge n$

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  • $\begingroup$ Is order of the $x_i$'s important? $\endgroup$ – Dan Sep 14 '14 at 11:25
  • $\begingroup$ @Dan, yes its important $\endgroup$ – bistaumanga Sep 14 '14 at 11:29
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This is given by the formula $$CC_k^n = {n+k-1 \choose n}$$

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  • $\begingroup$ so is this problem equivalent to distributing $n$ distinct balls into $k$ distinct boxes? $\endgroup$ – bistaumanga Sep 14 '14 at 11:58
  • $\begingroup$ $n$ similar balls into $k$ distinct boxes $\endgroup$ – Snufsan Sep 14 '14 at 12:03

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