Question

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:

• A. $201$
• B. $199$
• C. $101$
• D. $200$

Answer 1

The correct option is A

$201$

Explanation for correct option:

Finding the value of $n$:

It is given that, there are $n$ metro stations located in a city along a circular path. Each pair of the nearest station is connected by a blue line whereas all remaining are connected by a red line.

Also the number of red lines is equal to $99$ times the number of blue lines Therefore we have,

Number of blue lines $=$ Number of sides $=n$

Also, number of red lines $=$ number of diagonals$={}^{n}C_2-n$

Therefore, according to the question,

$$\begin{gathered} {}^{n}C_2-n=99n \\ \Rightarrow {}^{n}C_2=100n \\ \Rightarrow \frac{n\left(n-1\right)}{2}=100n \\ \Rightarrow \frac{n-1}{2}=100 \\ \Rightarrow n=200+1 \\ \Rightarrow n=201 \end{gathered}$$

Therefore, the number of metro stations $n$ is equal to $201$

Hence, the correct answer is option (A).