Complex Numbers
Complex Numbers
Allen Star Batch
Grade 11
Question:
If $z_1, z_2$ and $z_3$ are three complex numbers such that $|z_1| = |z_2| = |z_3| = 1$, then $|z_1 - z_2|^2 + |z_2 - z_3|^2 + |z_3 - z_1|^2$ is strictly less than
Step-by-Step Solution
Key Concept: Partition the selected digits into two groups and account for the symmetry of arrangements.
The total selections of 7 digits from digits 1–9 is $\binom{9}{7}$. The digits excluding 9 (the greatest) number 6, which can be divided into two groups of 3 each. These can be distributed as $\binom{6}{3} \times \binom{6}{3} \times \frac{1}{2!}$. However, the 3 digits on one side can move to the other, giving $\binom{9}{7} \cdot \binom{6}{3} \cdot \frac{2!}{2!} = \binom{9}{7} \cdot \binom{6}{3} \cdot \binom{6}{3}$.
Correct Answer: 3,4