jmc

We know that $\left(\genfrac{}{}{0ex}{}{19}{12}\right)=\left(\genfrac{}{}{0ex}{}{18}{11}\right)+\left(\genfrac{}{}{0ex}{}{18}{12}\right)$ from Pascal's identity. Solving for $\left(\genfrac{}{}{0ex}{}{18}{12}\right)$ and substituting the value that we have for $\left(\genfrac{}{}{0ex}{}{19}{12}\right)$ gives us $\left(\genfrac{}{}{0ex}{}{18}{12}\right)=50388-\left(\genfrac{}{}{0ex}{}{18}{11}\right)$. Once again using Pascal's identity, we know that $\left(\genfrac{}{}{0ex}{}{18}{11}\right)=\left(\genfrac{}{}{0ex}{}{17}{11}\right)+\left(\genfrac{}{}{0ex}{}{17}{10}\right)$. Substituting the values that we have for the terms on the right side gives us $\left(\genfrac{}{}{0ex}{}{18}{11}\right)=31824$ and substituting that into our expression for $\left(\genfrac{}{}{0ex}{}{18}{12}\right)$ gives us $\left(\genfrac{}{}{0ex}{}{18}{12}\right)=50388-31824=\boxed{18564}$.