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Permutations and Combinations question

2020 · 2 Sep · Shift 2 · Q27
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  5. /2020 · 2 Sep · Shift 2 · Q27

Permutations and Combinations question

2020 · 2 Sep · Shift 2 · Q27

JEE MainMathematicsPermutations and CombinationsMCQ+4 / −1
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. A
    201
  2. B
    199
  3. C
    101
  4. D
    200
View written solutionFree

Correct answer: A

  1. Count the blue lines

Since the nnn stations lie on a circle, each station is connected to its two nearest neighbors. The blue lines are exactly the sides of the nnn-gon formed by the stations.

Hence, the number of blue lines is n.n.n.

  1. Count the total number of straight tracks

Every pair of stations is connected by a straight track. Therefore, the total number of tracks is (n2)=n(n−1)2.\binom{n}{2} = \frac{n(n-1)}{2}.(2n​)=2n(n−1)​.

  1. Count the red lines

Red lines are all the remaining connections except the blue ones. So, red lines=(n2)−n=n(n−1)2−n.\text{red lines} = \binom{n}{2} - n = \frac{n(n-1)}{2} - n.red lines=(2n​)−n=2n(n−1)​−n.

Simplify: red lines=n(n−1)−2n2=n(n−3)2.\text{red lines} = \frac{n(n-1)-2n}{2} = \frac{n(n-3)}{2}.red lines=2n(n−1)−2n​=2n(n−3)​.

  1. Use the given condition

The number of red lines is 999999 times the number of blue lines: n(n−3)2=99n.\frac{n(n-3)}{2} = 99n.2n(n−3)​=99n.

Since n>2n>2n>2, we can divide by nnn: n−32=99.\frac{n-3}{2} = 99.2n−3​=99.

So, n−3=198n-3 = 198n−3=198 n=201.n = 201.n=201.

  1. Check the options

The value is n=201,n=201,n=201, which corresponds to Option A.

  1. Comparison with stored answer

Stored correct answer: A

Our derived answer: A

So, the answer agrees with the stored answer.

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