Final answer:
After applying combinatorial principles to the problem, the correct answer is found to be 201 Metro stations, where the number of red lines is 99 times the number of blue lines.
Step-by-step explanation:
The question is about determining the number of Metro stations, denoted as n, in a city, based on given conditions about connections between the stations.
Since each pair of nearest stations (forming an n-gon) is connected by a blue line, there are n blue lines. For red lines, they connect all other pairs of stations, so we have:
The total number of ways to connect the stations with straight lines is given by the formula for combinations: C(n, 2), which represents the number of ways to select 2 stations out of n to draw a line.
Since the number of red lines equals 99 times the number of blue lines, and there are n blue lines, the number of red lines is 99n.
The combination formula for C(n, 2) equals n(n - 1) / 2. Setting this equal to 99n + n (total number of lines), we have:
n(n - 1) / 2 = 99n + n
n(n - 1) = 2(99n + n) = 200n
n - 1 = 200
n = 201
Therefore, the number of Metro stations n is 201, which corresponds to answer option (3).