If $z$ is a complex number such that $\left|z + \frac{1}{z}\right| = 2$ then minimum value of $|z|$ is ____.
Step-by-Step Solution
Key Concept: Use stars and bars for distribution, then multiply by $200!$ to account for permutations of distinguishable persons.
Let $x_1, x_2, \ldots, x_7$ be the number of persons entering through gates $A, B, C, \ldots, G$ respectively, where $x_1 + x_2 + \cdots + x_7 = 200$. The number of ways to distribute into gates is $^{200+7-1}C_{7-1} = ^{206}C_6 = ^{206}C_{200}$. Since the arrangement of the 200 persons also matters, total ways = $^{206}C_{200} × 200! = ^{206}P_{200}$. With $n = 206$ and $r = 200$, we get $n - r = 6$, so the answer is $^{206}P_6$.
Correct Answer: 0.414