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8 circles of diameter 1 cm are kept in a row, each circle touching its

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8 circles of diameter 1 cm are kept in a row, each circle touching its  [#permalink]

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New post Updated on: 13 Nov 2015, 23:56
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8 circles of diameter 1 cm are kept in a row, each circle touching its neighbours. How many paths of length 4π are possible to go from A(0,0) to B(8,0) if you are not allowed to traverse back?

A. 128
B. 512
C. 256
D. 240
E. 300

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Originally posted by rsaahil90 on 13 Nov 2015, 23:19.
Last edited by Bunuel on 13 Nov 2015, 23:56, edited 1 time in total.
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Re: 8 circles of diameter 1 cm are kept in a row, each circle touching its  [#permalink]

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New post 14 Nov 2015, 02:33
rsaahil90 wrote:
8 circles of diameter 1 cm are kept in a row, each circle touching its neighbours. How many paths of length 4π are possible to go from A(0,0) to B(8,0) if you are not allowed to traverse back?

A. 128
B. 512
C. 256
D. 240
E. 300


There are two ways to reach from start to first tangential point

There are again 2 ways to reach from first tangential point to second tangential point of circles

There are always 2 ways to reach from one tangential to other in case of other circles as well

hence total ways to reach from start to end point = 2^8 = 256

Answer: option C
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Re: 8 circles of diameter 1 cm are kept in a row, each circle touching its  [#permalink]

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New post 14 Nov 2015, 07:28
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rsaahil90 wrote:
8 circles of diameter 1 cm are kept in a row, each circle touching its neighbours. How many paths of length 4π are possible to go from A(0,0) to B(8,0) if you are not allowed to traverse back?

A. 128
B. 512
C. 256
D. 240
E. 300



Circumference of each circle is => 2*pi*r => 2*pi*1/2 => pi
if we divide each circle into two semi circles, we will have 16 semi circles with each circumference as pi/2.

Every circle has two choice of paths, upper semi circle or lower semi circle.

From each circle, there are 2 ways to select one semi circle. SInce there are 8 semi circles, the number of ways to select is 2^8 => 256

Answer C
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Re: 8 circles of diameter 1 cm are kept in a row, each circle touching its  [#permalink]

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New post 04 Dec 2015, 00:00
Hi All,

This question is poorly written, so even though the basic concepts (geometry and permutations) will show up on Test Day, there are far more realistic practice resources that you could be using to learn those concepts.

Among the non GMAT-like details, the description of the circles should be "tangent to one another along a straight line" and the prompt refers to graphing co-ordinates, but doesn't establish that we're actually dealing with the XY co-ordinate plane.

In real basic terms, each of the 8 circles offers 2 options to get 'around it', which means that there are 2^8 ways to get from the 'beginning' to the 'end.' GMAT question writers are far more rigorous and detail-oriented about how they craft their questions and you should be sure that you're working with resources that train you to face the terminology and concepts that will appear on Test Day.

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Re: 8 circles of diameter 1 cm are kept in a row, each circle touching its  [#permalink]

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New post 01 Aug 2019, 05:45
EMPOWERgmatRichC wrote:
Hi All,

This question is poorly written, so even though the basic concepts (geometry and permutations) will show up on Test Day, there are far more realistic practice resources that you could be using to learn those concepts.

Among the non GMAT-like details, the description of the circles should be "tangent to one another along a straight line" and the prompt refers to graphing co-ordinates, but doesn't establish that we're actually dealing with the XY co-ordinate plane.

In real basic terms, each of the 8 circles offers 2 options to get 'around it', which means that there are 2^8 ways to get from the 'beginning' to the 'end.' GMAT question writers are far more rigorous and detail-oriented about how they craft their questions and you should be sure that you're working with resources that train you to face the terminology and concepts that will appear on Test Day.

GMAT assassins aren't born, they're made,
Rich

Hello,
In what way does the length of 4pi influence our calculations?
The existence of 2 ways per circle ^8 seams too easy as an answer and I am wondering how the answer would change if the length of 4pi would be changed.
Thank you
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Re: 8 circles of diameter 1 cm are kept in a row, each circle touching its  [#permalink]

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New post 01 Aug 2019, 16:30
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Hi Luca1111111111111,

The prompt tells us that we are "not allowed to traverse back", which implies that we cannot "trace our steps" when traveling along the circles. In simple terms, you cannot 'go backwards' or go around a particular circle multiple times. By telling us this, the length of the circles actually ends up having NOTHING to do with the answer to the question - we can only travel on 1/2 of each circle and then we have to transfer to the next circle. We're going from point A to point B, so there are 2^8 possible paths to take.

As it stands, since each circle has a diameter of 1 cm, each has a circumference of π cm. When you travel around 8 "halves" (which measure (1/2)π cm each), you'll travel 4π cm of total distance. If you change the total distance traveled, then you also have to change other aspects of the question to account for that new distance.

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Re: 8 circles of diameter 1 cm are kept in a row, each circle touching its  [#permalink]

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New post 01 Aug 2019, 22:03
EMPOWERgmatRichC wrote:
Hi Luca1111111111111,

The prompt tells us that we are "not allowed to traverse back", which implies that we cannot "trace our steps" when traveling along the circles. In simple terms, you cannot 'go backwards' or go around a particular circle multiple times. By telling us this, the length of the circles actually ends up having NOTHING to do with the answer to the question - we can only travel on 1/2 of each circle and then we have to transfer to the next circle. We're going from point A to point B, so there are 2^8 possible paths to take.

As it stands, since each circle has a diameter of 1 cm, each has a circumference of π cm. When you travel around 8 "halves" (which measure (1/2)π cm each), you'll travel 4π cm of total distance. If you change the total distance traveled, then you also have to change other aspects of the question to account for that new distance.

GMAT assassins aren't born, they're made,
Rich

Thank you very much - great explanation!
May I ask another question (hopefully not too stupid): The formula to calculate the a circle's circumference is 2πr. Isn't the circumference of the circle 2π then and of the halves just π?
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Re: 8 circles of diameter 1 cm are kept in a row, each circle touching its  [#permalink]

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New post 02 Aug 2019, 13:25
Hi Luca1111111111111,

We're told that each circle has a DIAMETER of 1 cm, which means that each RADIUS is 1/2 cm. You are correct that the formula for a circle's circumference = 2π(radius). Here, that would be 2π (1/2) = π... so each half-circumference would be (1/2)π.

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Re: 8 circles of diameter 1 cm are kept in a row, each circle touching its  [#permalink]

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New post 16 Aug 2019, 20:09
Doesn't the length of 4π limit the start to the first 5 circles?

From the first circle there are 2^4 options
From the second circle there are 2^4 options
Up to the fifth circle having 2^4 options (the circles thereafter can't have lines of 4π length without going backwards)

5·2^4=90

How is the above thinking wrong?
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8 circles of diameter 1 cm are kept in a row, each circle touching its  [#permalink]

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New post 16 Aug 2019, 23:36
rsaahil90 wrote:
8 circles of diameter 1 cm are kept in a row, each circle touching its neighbours. How many paths of length 4π are possible to go from A(0,0) to B(8,0) if you are not allowed to traverse back?

A. 128
B. 512
C. 256
D. 240
E. 300


Given: 8 circles of diameter 1 cm are kept in a row, each circle touching its neighbours.

Asked: How many paths of length 4π are possible to go from A(0,0) to B(8,0) if you are not allowed to traverse back?

There are 8 circles touching one another with circumference \(= \pi\)
There are 2 options for each circle = upper semi-circle or lower semi-circle of lengths \(= \pi/2\)
So total ways (options) for path of length \(( 8 * \pi/2 = 4 \pi) = 2^8 = 256\)

IMO C
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Re: 8 circles of diameter 1 cm are kept in a row, each circle touching its  [#permalink]

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New post 16 Aug 2019, 23:40
philipssonicare wrote:
Doesn't the length of 4π limit the start to the first 5 circles?

From the first circle there are 2^4 options
From the second circle there are 2^4 options
Up to the fifth circle having 2^4 options (the circles thereafter can't have lines of 4π length without going backwards)

5·2^4=90

How is the above thinking wrong?


There are 8 semi-circles of length (\pi/2) making a total path length of \(8 * \pi/2 = 4 \pi\)

So there is no other option than to choose either a upper semi-circle or a lower semi-circle for each circle in the path without going backwards.
Total ways = \(2^8 = 256\)
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Re: 8 circles of diameter 1 cm are kept in a row, each circle touching its   [#permalink] 16 Aug 2019, 23:40
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