The value of $4\alpha\left(\beta^4 - \alpha^4\right)$, if $\alpha + i\beta, \beta \neq 0$ is a root of $z^5 = -1$, is ____.
Step-by-Step Solution
Key Concept: Use generating functions with the constraint that each die shows 1–6, then subtract from total outcomes.
Let $x_1, x_2, ..., x_6$ be die outcomes with $x_1 + x_2 + ... + x_6 \leq 17$. Introducing dummy variable $x_7$ makes $x_1 + x_2 + ... + x_6 + x_7 = 17$ where $1 \leq x_i \leq 6$ for $i = 1,...,6$ and $0 \leq x_7$. The generating function coefficient of $x^{17}$ in $(x+x^2+...+x^6)^6(1+x+x^2+...)$ equals the coefficient in $(1-x^6)^6(1-x)^{-7}$. This equals $\binom{17}{11} - 6\binom{11}{5} = 46656 - 12376 - 6 \times 642 = 31508$. Required ways for sum greater than 17: $6^6 - 31508 = 46656 - 31508 = 3150$.
Correct Answer: 1