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 the blue line, whereas all remaining pairs of stations are connected by the red line. If the number of red lines is $99$ times the number of blue lines, then the value of $n$ is:
1. 201
2. 199
3. 101
4. 200

Answer 1

Hint: In this question, we are given that there are $n$ metro stations in a city and each pair of metro stations is connected by a straight track. Write the number of blue lines and red lines. Solve further using the condition that a number of red lines are $99$ times the number of blue lines to know the value of $n$.

Formula Used:
Combination formula –
${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}},n! = n\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right).......$

Complete step by step Solution:
Given that,
There are $n$ metro stations in a city that are located along a circular path and each pair of stations are connected by a straight track only.
Here, the number of sides is $n$
Therefore, the number of blue lines $ = n$
Now there are remaining pairs of metro stations (except those which are connected to the blue line) that are connected by blue lines, and they are equal to a number of diagonals.
$ \Rightarrow $Number of red lines are ${}^n{C_2} - n$
According to the question,
Number of red lines is $99$ times the number of blue lines
It implies that,
${}^n{C_2} - n = 99n$
Using combination formula,
$\dfrac{{n!}}{{2!\left( {n - 2} \right)!}} - n = 99n$
$\dfrac{{n\left( {n - 1} \right)\left( {n - 2} \right)!}}{{\left( {2 \times 1} \right)\left( {n - 2} \right)!}} - n = 99n$
$\dfrac{{n\left( {n - 1} \right)}}{2} - n = 99n$
$\dfrac{{{n^2} - n - 2n}}{2} = 99n$
${n^2} - 3n = 198n$
$n - 3 = 198$
$n = 201$

Hence, the correct option is 1.

Note: To solve such a question, students must have good understanding skills. As in this question, it is written that pair of a metro stations is connected by a straight track which means a number of red lines will be the combination of $2$ from $n$. Also, $n$ is subtracted because we don’t have to count blue lines in red and in ${}^n{C_2}$, $n$ is there.