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All together we get that the number of ways to distribute 10 cookies to 4 kids without giving any kid more than 2 cookies is:

\begin{equation*} {13 \choose 3} - \left[{4 \choose 1}{10 \choose 3} - {4 \choose 2}{7 \choose 3} + {4\choose 3}{4\choose 3}\right] \end{equation*}

which is

\begin{equation*} 286 - [480 - 210 + 16] = 0\text{.} \end{equation*}
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