If 3 a + 3 b = 3 c , then the simplest value of ( a + b - c )^3 + 27abc is?

Answer: Option D

$$\begin{array}{rl}& \text{root}3\text{of}a+\text{root}3\text{of}b=\text{root}3\text{of}c\\ & \text{Take cube both sides}\\ & {\left(\text{root}3\text{of}a+\text{root}3\text{of}b\right)}^3={\left(\text{root}3\text{of}c\right)}^3\\ & \Rightarrow a+b+3a^{\frac{1}{3}}{b}^{\frac{1}{3}}\left(\text{root}3\text{of}a+\text{root}3\text{of}b\right)=c\\ & \Rightarrow a+b+3a^{\frac{1}{3}}{b}^{\frac{1}{3}}{c}^{\frac{1}{3}}=c\\ & \Rightarrow a+b-c=-3a^{\frac{1}{3}}{b}^{\frac{1}{3}}{c}^{\frac{1}{3}}\\ & \text{Again take cube both sides}\\ & \Rightarrow {\left(a+b-c\right)}^3=-27abc\\ & \Rightarrow {\left(a+b-c\right)}^3+27abc=0\\ & \\ & \mathbf{Alternate}:\\ & \text{Put value of }\\ & a=0\\ & b=1\\ & c=1\\ & \text{Value of }{\left(a+b-c\right)}^3+27abc\\ & ={\left(0+1-1\right)}^3+27\times 0\times 1\times 1\\ & =0\end{array}$$