$\begin{align}& \\ & \,\,\,\,\left( \frac{2\sqrt{6}}{3}-\frac{2\sqrt{3}-\sqrt{6}}{\sqrt{8}-2} \right).\frac{1}{\sqrt{6}} \\ & =\left( \frac{2\sqrt{2}.\sqrt{3}}{\sqrt{3}.\sqrt{3}}-\frac{2\sqrt{3}-\sqrt{2}.\sqrt{3}}{2\sqrt{2}-2} \right).\frac{1}{\sqrt{6}} \\ & =\left( \frac{2\sqrt{2}}{\sqrt{3}}-\frac{\sqrt{2}.\sqrt{3}\left( \sqrt{2}-1 \right)}{2(\sqrt{2}-1)} \right).\frac{1}{\sqrt{6}} \\ & =\left( \frac{2\sqrt{2}}{\sqrt{3}}-\frac{\sqrt{3}}{\sqrt{2}} \right).\frac{1}{\sqrt{6}} \\&=\frac{2.\sqrt{2}.\sqrt{2}-\sqrt{3}.\sqrt{3}}{\sqrt{2}.\sqrt{3}}.\frac{1}{\sqrt{2}.\sqrt{3}} \\
& =\frac{4-3}{2.3} \\ & =\frac{1}{6} \\ \end{align}$