The value of $$\frac{3\sqrt{2}}{\sqrt{3} + \sqrt{6}} - \frac{4\sqrt{3}}{\sqrt{6}+\sqrt{2}} + \frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}} $$ is

Expression : $$\frac{3\sqrt{2}}{\sqrt{3} + \sqrt{6}} - \frac{4\sqrt{3}}{\sqrt{6}+\sqrt{2}} + \frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}} $$

= $$\frac{3\sqrt2(\sqrt6+\sqrt2)(\sqrt3+\sqrt2)-4\sqrt3(\sqrt3+\sqrt6)(\sqrt3+\sqrt2)+\sqrt6(\sqrt3+\sqrt6)(\sqrt6+\sqrt2)}{(\sqrt3+\sqrt6)(\sqrt6+\sqrt2)(\sqrt3+\sqrt2)}$$

= $$\frac{1}{(\sqrt3+\sqrt6)(\sqrt6+\sqrt2)(\sqrt3+\sqrt2)}\times$$ $$[3\sqrt2(\sqrt{18}+\sqrt{12}+\sqrt6+\sqrt4)-4\sqrt3(\sqrt9+\sqrt6+\sqrt{18}+\sqrt{12}) + \sqrt6(\sqrt{18}+ \sqrt6+\sqrt{36}+\sqrt{12})]$$

= $$\frac{1}{(\sqrt3+\sqrt6)(\sqrt6+\sqrt2)(\sqrt3+\sqrt2)}\times$$ $$[3\sqrt2(3\sqrt2+2\sqrt3+\sqrt6+2)-4\sqrt3(3+\sqrt6+3\sqrt2+2\sqrt3)+ \sqrt6(3\sqrt2+\sqrt6+6+2\sqrt3)]$$

= $$\frac{1}{(\sqrt3+\sqrt6)(\sqrt6+\sqrt2)(\sqrt3+\sqrt2)}\times$$ $$[(18+6\sqrt6+6\sqrt3+6\sqrt2)+(-12\sqrt3-12\sqrt2-12\sqrt6-24)+ (6\sqrt3+6+6\sqrt6+6\sqrt2)]$$

= $$\frac{1}{(\sqrt3+\sqrt6)(\sqrt6+\sqrt2)(\sqrt3+\sqrt2)}\times$$ $$[(18+6-24)+(6\sqrt2-12\sqrt2+6\sqrt2)+(6\sqrt3-12\sqrt3+6\sqrt3)+(6\sqrt6-12\sqrt6+6\sqrt6)]$$

= $$\frac{1}{(\sqrt3+\sqrt6)(\sqrt6+\sqrt2)(\sqrt3+\sqrt2)}\times0=0$$

=> Ans - (B)