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With $\ds{N \in {\mathbb N}}$:
\begin{align}
&\color{#c00000}{\sum_{x_{1} = 0}^{\infty}\ldots\sum_{x_{6} = 0}^{\infty}
\delta_{x_{1} + \cdots + x_{6},N}}
=\sum_{x_{1} = 0}^{\infty}\ldots\sum_{x_{6} = 0}^{\infty}\oint_{\verts{z}\ =\ 1}
{1 \over z^{-x_{1} - \cdots - x_{6} + N + 1}}\,{\dd z \over 2\pi\ic}
\\[3mm]&=\oint_{\verts{z}\ =\ 1}{1 \over z^{N + 1}}
\pars{\sum_{x = 0}^{\infty}z^{x}}^{6}\,{\dd z \over 2\pi\ic}
=\oint_{\verts{z}\ =\ 1}{1 \over z^{N + 1}}
{1 \over \pars{1 - z}^{6}}\,{\dd z \over 2\pi\ic}
\\[3mm]&=\oint_{\verts{z}\ =\ 1}{1 \over z^{N + 1}}
\sum_{n = 0}^{\infty}{-6 \choose n}\pars{-1}^{n}z^{n}\,{\dd z \over 2\pi\ic}
=\sum_{n = 0}^{\infty}{-6 \choose n}\pars{-1}^{n}\
\overbrace{\oint_{\verts{z}\ =\ 1}{z^{n} \over z^{N + 1}}
\,{\dd z \over 2\pi\ic}}^{\ds{=\ \delta_{nN}}}
\\[3mm]&=\pars{-1}^{N}{-6 \choose N} = \pars{-1}^{N}\bracks{\pars{-1}^{N}{-\bracks{-6} + N - 1 \choose N}}
=\color{#c00000}{{N + 5 \choose 5}}
\end{align}
\begin{align}
&\color{#c00000}{\sum_{N = 0}^{9}{N + 5 \choose 5}}
=\sum_{N = 0}^{9}\oint_{\verts{z}\ =\ 1}{\pars{1 + z}^{N + 5} \over z^{6}}
\,{\dd z \over 2\pi\ic}
=\oint_{\verts{z}\ =\ 1}{1 \over z^{6}}\sum_{N = 0}^{9}\pars{1 + z}^{N + 5}
\,{\dd z \over 2\pi\ic}
\\[3mm]&=\oint_{\verts{z}\ =\ 1}{1 \over z^{6}}\,
{\pars{1 + z}^{5}\bracks{\pars{1 + z}^{10} - 1} \over \pars{1 + z} - 1}
\,{\dd z \over 2\pi\ic}
\\[3mm]&=\overbrace{\oint_{\verts{z}\ =\ 1}{\pars{1 + z}^{15} \over z^{7}}\,{\dd z \over 2\pi\ic}}^{\ds{=\ {15 \choose 6}}}\
-\
\overbrace{\oint_{\verts{z}\ =\ 1}{\pars{1 + z}^{5} \over z^{7}}\,{\dd z \over 2\pi\ic}}^{\ds{=\ 0}}\ =\ {15 \choose 6} = \color{#00f}{\large 5005}
\end{align}
