If $\omega \neq 1$ is a cube root of unity and $z$ is a complex number such that $|z| = 1$ then $\left|\frac{2 + 3\omega + 4z\omega^2}{4\omega + 3\omega^2 z + 2z}\right| = $ ____.
Step-by-Step Solution
Key Concept: Systematically enumerate all valid distributions satisfying row constraints, then count arrangements for each case.
The problem distributes $x$ identical objects into three rows with constraints. Case I: first row gets $1x$, second gets $4x$, third gets $1x$ giving $2 × 1 × 2 = 4$ ways. Case II: distribution $(2x, 3x, 1x)$ gives $1 × ^4C_3 × 2 = 8$ ways. Case III: distribution $(1x, 3x, 2x)$ gives $2 × ^4C_3 × 1 = 8$ ways. Case IV: distribution $(2x, 2x, 2x)$ gives $1 × ^4C_2 × 1 = 6$ ways. Total = $4 + 8 + 8 + 6 = 26$ ways.
Correct Answer: 1