JEE Main 2020 - 2 Sep (Evening)

LEVELJEE Main

Animated Solution for Mathematics - Permutations and Combinations: 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:

Q: Can I use the formula for the number of diagonals directly?

Absolutely! The number of diagonals in an $n$-sided polygon is given by $\frac{n(n-3)}{2}$. If you set this equal to $99n$, you get $\frac{n(n-3)}{2} = 99n$. Since $n > 2$, you can divide by $n$ to get $\frac{n-3}{2} = 99$, which leads to $n-3 = 198$, or $n = 201$. Both methods lead to the same beautiful result.

Q: Is it mathematically valid to cancel $n$ from both sides of the equation?

Yes, it is perfectly valid here. The problem states that $n > 2$, which guarantees that $n$ is not zero. In algebra, you can divide by any variable as long as you are certain it is non-zero. Since $n$ represents the number of stations, it must be a positive integer, so dividing by $n$ is safe and simplifies the equation significantly.

Select Answer:

Visualized Solution

Visualizing the Circular Path

Let $n$ be the number of metro stations arranged on a circular path.
Each pair of stations is connected by a straight track.

Identifying Blue Lines

Blue lines connect nearest neighbor stations.
In a circular arrangement of $n$ points, there are exactly $n$ adjacent pairs.
Therefore, Number of blue lines = $n$ .

Total Number of Tracks

Total number of straight tracks is the number of ways to choose $2$ stations from $n$ .
Total tracks = $^{n} C_{2} = \frac{n ( n - 1 )}{2}$ .

Defining Red Lines

Red lines connect all non-adjacent pairs of stations.
Number of red lines = Total tracks $-$ Number of blue lines.
Number of red lines = $^{n} C_{2} - n$ .

Formulating the Equation

Given: Number of red lines = $99 \times$ Number of blue lines.
Substitute the expressions: $^{n} C_{2} - n = 99 n$ .

Rearranging the Equation

Move $- n$ to the right side: $^{n} C_{2} = 99 n + n$ .
Simplify: $^{n} C_{2} = 100 n$ .

Expanding $^{n} C_{2}$

Substitute the formula for $^{n} C_{2}$ :
$\frac{n ( n - 1 )}{2} = 100 n$ .

Canceling $n$

Since $n > 2$ , $n \neq = 0$ .
Divide both sides by $n$ :
$\frac{n - 1}{2} = 100$ .

Solving for $n$

Multiply both sides by $2$ : $n - 1 = 200$ .
Add $1$ to both sides: $n = 201$ .

Final Conclusion

The number of stations $n$ is 201.
Key Takeaway: In an $n$ -gon, there are $n$ sides and $\frac{n ( n - 3 )}{2}$ diagonals.
Final Answer: 201

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The Sigma Insight: Combinations and Selection

Solution Diagram

Analyzing the Geometry of the Metro System

Imagine

n

metro stations arranged as vertices of a regular

n

-sided polygon. The blue lines represent the sides of this polygon, connecting adjacent stations.

Since a polygon with

n

vertices has exactly

n

sides, the number of blue lines is:

Blue lines = n

Calculating Total Connections

To find the total number of tracks, we must calculate the number of ways to choose

2

stations out of

n

available vertices. This is given by the combination formula

^{n} C_{2}

The total number of tracks is:

Total tracks = \frac{n ( n - 1 )}{2}

The red lines represent the diagonals of the polygon, which are all tracks that are not sides. We find this by subtracting the blue lines from the total tracks:

Red lines = \frac{n ( n - 1 )}{2} - n

The Master Equation

We are given the condition that the number of red lines is

99

times the number of blue lines. We can express this relationship as:

\frac{n ( n - 1 )}{2} - n = 99 n

To solve for

n

, first add

n

to both sides to isolate the total number of tracks:

\frac{n ( n - 1 )}{2} = 100 n

Final Calculation

Since the problem implies a polygon exists, we know

n > 2

, meaning $n eq 0$. We can safely divide both sides of the equation by

n

\frac{n - 1}{2} = 100

Multiplying both sides by

2

yields:

n - 1 = 200

Adding

1

to both sides, we find the total number of stations: $n = 201$

Common Traps & FAQs

Why is the number of blue lines equal to

n

Think of the stations as vertices of a polygon. In any polygon with

n

vertices, there are exactly

n

sides connecting adjacent vertices. Since the blue lines connect only the nearest neighbors, they represent exactly these

n

sides.

Can I use the formula for the number of diagonals directly?

Absolutely! The number of diagonals in an

n

-sided polygon is given by

\frac{n ( n - 3 )}{2}

. If you set this equal to

99 n

, you get

\frac{n ( n - 3 )}{2} = 99 n

. Since

n > 2

, you can divide by

n

to get

\frac{n - 3}{2} = 99

, which leads to

n - 3 = 198

, or

n = 201

. Both methods lead to the same beautiful result.

Is it mathematically valid to cancel

n

from both sides of the equation?

Yes, it is perfectly valid here. The problem states that

n > 2

, which guarantees that

n

is not zero. In algebra, you can divide by any variable as long as you are certain it is non-zero. Since

n

represents the number of stations, it must be a positive integer, so dividing by

n

is safe and simplifies the equation significantly.

Select Answer:

Visualized Solution

Visualizing the Circular Path

Identifying Blue Lines

Total Number of Tracks

Defining Red Lines

Formulating the Equation

Rearranging the Equation

Expanding $^{n} C_{2}$

Canceling $n$

Solving for $n$

Final Conclusion

The Sigma Insight: Combinations and Selection

Analyzing the Geometry of the Metro System

Calculating Total Connections

The Master Equation

Final Calculation

Similar Questions

Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number beams is :

The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is ______.

If for some $m, n$ ; $^{6} C_{m} + 2 (^{6} C_{m + 1}) +^{6} C_{m + 2} >^{8} C_{3}$ and $^{n - 1} P_{3} :^{n} P_{4} = 1 : 8$ , then $^{n} P_{m + 1} +^{n + 1} C_{m}$ is equal to

Let $0 \leq r \leq n$ . If $^{n + 1} C_{r + 1} :^{n} C_{r} :^{n - 1} C_{r - 1} = 55 : 35 : 21$ , then $2 n + 5 r$ is equal to:

For $n \geq 2$ , let $S_{n}$ denote the set of all subsets of ${1, 2, \dots, n}$ with no two consecutive numbers. For example ${1, 3, 5} \in S_{6}$ , but ${1, 2, 4} \in / S_{6}$ . Then $n (S_{5})$ is equal to ________

There are $m$ men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84, then the value of $m$ is :

Let S denote the set of 4-digit numbers $ab c d$ such that $a > b > c > d$ and P denote the set of 5-digit numbers having product of its digits equal to 20. Then $n (S) + n (P)$ is equal to ......... .

These are 10 points in a plane, out of these 6 are collinear, if $N$ is the number of triangles formed by joining these points. then:

Let $T_{n}$ be the number of all possible triangles formed by joining vertices of an $n$ -sided regular polygon. If $T_{n + 1} - T_{n} = 10$ , then the value of $n$ is :

Select Answer:

Visualized Solution

Visualizing the Circular Path

Identifying Blue Lines

Total Number of Tracks

Defining Red Lines

Formulating the Equation

Rearranging the Equation

Expanding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } nC2​

Canceling .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n

Final Conclusion

The Sigma Insight: Combinations and Selection

Analyzing the Geometry of the Metro System

Calculating Total Connections

The Master Equation

Final Calculation

Similar Questions

Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number beams is :

The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is ______.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If for some m,n; 6Cm​+2(6Cm+1​)+6Cm+2​>8C3​ and n−1P3​:nP4​=1:8, then nPm+1​+n+1Cm​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let 0≤r≤n. If n+1Cr+1​:nCr​:n−1Cr−1​=55:35:21, then 2n+5r is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } These are 10 points in a plane, out of these 6 are collinear, if N is the number of triangles formed by joining these points. then:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let Tn​ be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If Tn+1​−Tn​=10, then the value of n is :

Expanding $^{n} C_{2}$

Canceling $n$

Solving for $n$

If for some $m, n$ ; $^{6} C_{m} + 2 (^{6} C_{m + 1}) +^{6} C_{m + 2} >^{8} C_{3}$ and $^{n - 1} P_{3} :^{n} P_{4} = 1 : 8$ , then $^{n} P_{m + 1} +^{n + 1} C_{m}$ is equal to

Let $0 \leq r \leq n$ . If $^{n + 1} C_{r + 1} :^{n} C_{r} :^{n - 1} C_{r - 1} = 55 : 35 : 21$ , then $2 n + 5 r$ is equal to:

These are 10 points in a plane, out of these 6 are collinear, if $N$ is the number of triangles formed by joining these points. then:

Let $T_{n}$ be the number of all possible triangles formed by joining vertices of an $n$ -sided regular polygon. If $T_{n + 1} - T_{n} = 10$ , then the value of $n$ is :