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For $n\in \mathbb{N}$ and some prime $p$, consider $(\mathbb{F}_p)^n$. Is it known how many weak compositions $$x_1+x_2+\ldots +x_n\equiv 0 \pmod p$$ in $\mathbb{F}_p$ there are, where $(x_1, \ldots, x_n)\in (\mathbb{F}_p)^n$?

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Hint: We can pick $x_1$ to $x_{n-1}$ arbitrarily, and then $x_n$ is determined.

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