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Let n > 2 be an integer. Suppose that there are n Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number of red lines is 99 times the number of blue lines, then the value of n is:-
- 199
- 101
- 201
- 200
Correct answer: 201
Solution
In a circular arrangement of n stations, there are n blue lines connecting each pair of nearest stations. The total number of connections (lines) between all pairs of stations is given by the combination formula C(n, 2), which equals n(n-1)/2. The number of red lines is then the total lines minus the blue lines, leading to the equation (n(n-1)/2 - n) = 99n. Solving this equation results in n = 201.
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