If $|z| = 2$ and $\frac{z_1 - z_3}{z_2 - z_3} = \frac{z - 2}{z + 2}$, then $z_1, z_2, z_3$ will be vertices of a/an:
Step-by-Step Solution
Key Concept: Use multinomial coefficients to count arrangements within each case, then multiply by the number of ways to distribute distinct items.
The problem involves distributing objects into categories with constraints. Three cases are analyzed based on values of $P_1$, $P_2$, $P_3$: Case 1 gives $\frac{8!}{1!3!4!} \times 13 = 1680$, Case 2 gives $\frac{8!}{2!3!3!} \times \frac{13}{2} = 1680$, and Case 3 gives $\frac{8!}{2!2!4!} \times \frac{13}{2} = 1260$. The total is $1680 + 1680 + 1260 = 4620$. Therefore $K \times P_3 = 4620$, so $K = \frac{4620}{210} = 22$.
Correct Answer: 3