Step-by-Step Solution
Key Concept: Row and column operations combined with factorization can reduce a complex determinant to a simpler form involving a constraint on $x$, $y$, and $z$.
Multiply $R_1$ by $z$, $R_2$ by $y$, and $R_3$ by $x$, then divide the determinant by $xyz$. This transforms the determinant into $\frac{1}{xyz}\begin{vmatrix} z^3 + x & z^2y & xz^2 \\ zy^3 & y^3 + y & y^2 \\ zx^3 & yx^2 & x^3 + z \end{vmatrix} = 11$. Factoring out $(x^3 + y^3 + z^3 + 1)$ from the determinant and simplifying gives $x^3 + y^3 + z^3 = 10$. Therefore, the possible solutions are $(2, 1, 1)$, $(1, 2, 1)$, and $(1, 1, 2)$, giving us 3 solutions.
Correct Answer: 2