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Find a closed formula for the number of non-negative integral solutions for $x_1 + x_2 + \cdots +x_n = n$
Question: Find a closed formula for the number of non-negative integral solutions for $x_1 + x_2 + \cdots +x_n = n$ where $x_i >0$.
My guess: $S=\binom{n+k-1}{k-1}-\binom{n+k}{k}$, where $k=\max(x_1,x_2,\dots,x_n)$. But I don’t see how this leads to a closed formula.
A:
From your hint, we get $$\mathcal{S}=\binom{n+k-1}{k-1}-\binom{n+k}{k},$$
which is equivalent to
$$\mathcal{S}=\binom{n+k-1}{n-1}+\binom{n+k-2}{n-2}+\cdots+\binom{n+k-n+1}{0}.$$
Let
$$\mathcal{U}=\binom{n+k-1}{n-1}+\binom{n+k-2}{n-2}+\cdots+\binom{n+k-n+1}{0}.$$
We want to find a closed formula for $\mathcal{S}-\mathcal{U}$.
How many ways are there to choose $n-1$ integers among $1,2,\ldots,n+k-1$? I
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