Let n>2 be an integer. Suppose that there are


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 :

  1. (1) 201

  2. (2) 200

  3. (3) 101

  4. (4) 199

Correct Option: 1


(1) Number of two consecutive stations (Blue lines) $=n$ Number of two non-consecutive stations (Red lines)

$={ }^{n} C_{2}-n$

Now, according to the question, ${ }^{n} C_{2}-n=99 n$

$\Rightarrow \frac{n(n-1)}{2}-100 n=0 \Rightarrow n(n-1-200)=0$

$\Rightarrow n-1-200=0 \Rightarrow n=201$


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