Correct Answer: 0

Solution : $x + y = 36$
Formula used:
$a^3 + b^3 = (a + b)(a^2 - ab +  b^2)$
$⇒ x + y = 36$
$⇒ x + y - 36 = 0$--------(1)
$⇒ (x - 27) + (y - 9) = 0$
Let's consider $(x - 27) = a$ and $(y - 9) = b$
$a^3 + b^3 = (a + b)(a^2 + b^2 - ab)$ using the formula, substitute the value of a and b into the equation.
$(x - 27)^3 + (y - 9)^3= ( x - 27 + y - 9)((x - 27)^2 - (x - 27)(y - 9) + (y - 9)^2)$
$= (x - 27)^3 + (y - 9)^3 = ( x + y - 36)((x - 27)^2 - (x - 27)(y - 9) + (y - 9)^2)$
Substituting the value of $x + y - 36 = 0$ from equation (1).
$⇒ (x - 27)^3+ (y - 9)^3 = (0)((x - 27)^2 - (x - 27)(y - 9) + (y - 9)^2)$
$⇒ (x - 27)^3 + (y - 9)^3 = 0$
$\therefore$ The required value is 0.
Hence, the correct answer is 0.