The value of $$5\sqrt{3} + 7\sqrt{2} - \sqrt{6} - \frac{23}{\sqrt{2} + \sqrt{3} + \sqrt{6}}$$ is:

$$5\sqrt{3}+7\sqrt{2}-\sqrt{6}-\frac{23}{\sqrt{2}+\sqrt{3}+\sqrt{6}}=5\sqrt{3}+7\sqrt{2}-\sqrt{6}-\frac{23}{\sqrt{2}+\sqrt{3}+\sqrt{6}}\times\ \frac{\sqrt{2}+\sqrt{3}-\sqrt{6}}{\sqrt{2}+\sqrt{3}-\sqrt{6}}$$

$$=5\sqrt{3}+7\sqrt{2}-\sqrt{6}-\frac{23\left(\sqrt{2}+\sqrt{3}-\sqrt{6}\right)}{\left(\sqrt{2}+\sqrt{3}\right)^2-\left(\sqrt{6}\right)^2}\ $$

$$=5\sqrt{3}+7\sqrt{2}-\sqrt{6}-\frac{23\left(\sqrt{2}+\sqrt{3}-\sqrt{6}\right)}{2+3+2\sqrt{6}-6}\ $$

$$=5\sqrt{3}+7\sqrt{2}-\sqrt{6}-\frac{23\left(\sqrt{2}+\sqrt{3}-\sqrt{6}\right)}{2\sqrt{6}-1}\ $$

$$=5\sqrt{3}+7\sqrt{2}-\sqrt{6}-\frac{23\left(\sqrt{2}+\sqrt{3}-\sqrt{6}\right)}{2\sqrt{6}-1}\ \times\frac{2\sqrt{6}+1}{2\sqrt{6}+1}$$

$$=5\sqrt{3}+7\sqrt{2}-\sqrt{6}-\frac{23\left(\sqrt{2}+\sqrt{3}-\sqrt{6}\right)\left(2\sqrt{6}+1\right)}{\left(2\sqrt{6}\right)^2-1^2}\ $$

$$=5\sqrt{3}+7\sqrt{2}-\sqrt{6}-\frac{23\left(\sqrt{2}+\sqrt{3}-\sqrt{6}\right)\left(2\sqrt{6}+1\right)}{24-1}\ $$

$$=5\sqrt{3}+7\sqrt{2}-\sqrt{6}-\frac{23\left(2\sqrt{12}+\sqrt{2}-2\sqrt{18}+\sqrt{3}-12-\sqrt{6}\right)}{23}\ $$

$$=5\sqrt{3}+7\sqrt{2}-\sqrt{6}-4\sqrt{3}-\sqrt{2}-6\sqrt{2}-\sqrt{3}+12+\sqrt{6}\ $$

$$=12$$

Hence, the correct answer is Option D