After all that work we are finally ready to count. Each way to distribute cookies corresponds to a stars and bars chart with 7 stars and 3 bars. So there are 10 symbols, and we must choose 3 of them to be bars. Thus:
\begin{equation*}
\mbox{ There are } {10 \choose 3}\mbox{ ways to distribute 7 cookies to 4 kids}\text{.}
\end{equation*}