Given that $\binom{15}{8}=6435$, $\binom{16}{9}=11440$, and $\binom{16}{10}=8008$, find $\binom{15}{10}$.
We can use Pascal's identity $ \binom{n-1}{k-1}+\binom{n-1}{k}=\binom{n}{k}$ to find $\binom{15}{9}$.

\begin{align*}
\binom{15}{8}+\binom{15}{9}&=\binom{16}{9} \rightarrow \\
6435+\binom{15}{9}&=11440 \rightarrow \\
\binom{15}{9}&=5005
\end{align*}

We can use the identity again to find $\binom{15}{10}$.

\begin{align*}
\binom{15}{9}+\binom{15}{10}&=\binom{16}{10} \rightarrow \\
5005+\binom{15}{10}&=8008 \rightarrow \\
\binom{15}{10}&=3003
\end{align*}

Therefore, $\binom{15}{10}=\boxed{3003}$.