$\begin{array}{l}
\dfrac{1}{{1!2009!}} + \dfrac{1}{{2!.2008!}} + ... + \dfrac{1}{{2009!1!}}\\
= \dfrac{1}{{2010!}}.\left( {\dfrac{{2010!}}{{1.2010!}} + \dfrac{{2010!}}{{2!.2008!}} + .... + \dfrac{{2010!}}{{2009!1!}}} \right)\\
= \dfrac{1}{{2010!}}\left( {C_{2010}^1 + C_{2010}^2 + C_{2010}^3 + ... + C_{2010}^{2009}} \right)\\
= \dfrac{1}{{2010!}}\left( {C_{2010}^0 + C_{2010}^1 + C_{2010}^2 + C_{2010}^3 + ... + C_{2010}^{2010} - C_{2010}^0 - C_{2010}^{2010}} \right)\\
= \dfrac{1}{{2010!}}\left( {{2^{2010}} - 2} \right)\\
= \dfrac{{{2^{2010}} - 2}}{{2010!}}
\end{array}$
