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What are the number of solutions of $x_1+x_2+...x_n=k$ when $m+1\leq k\leq2m+1$ and $m\geq x_i \geq 0$?

I saw a solution the states it's $$CC^k_n-\begin{pmatrix} n\\1\end{pmatrix} \times CC^{k-(m+1)}_n$$

But I can't figure out what are these $\begin{pmatrix} n\\1\end{pmatrix} \times CC^{k-(m+1)}_n$ mean.

Thank you.

Evy
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    I have never seen the notation used either and doubt its accuracy... but if you are willing to use generating functions the answer is simply the coefficient of $x^k$ in the expansion of $(1+x+x^2+\dots+x^m)^n$. The condition that $m+1\leq k\leq 2m+1$ is entirely irrelevant. If you want a solution that doesn't use generating functions, then I recommend inclusion-exclusion based on the events of whether or not $x_i$ exceeds the value of $m$. – JMoravitz Apr 13 '19 at 13:17
  • This is a special case of https://math.stackexchange.com/q/553960/177399. The complicated formula there simplifies with the extra assumption on $k.$ – Mike Earnest Apr 13 '19 at 14:22

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