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# You are given an unlimited number of circles each of which

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You are given an unlimited number of circles each of which [#permalink]  11 Jan 2013, 00:20
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52% (02:31) correct 47% (01:56) wrong based on 51 sessions
You are given an unlimited number of circles each of which having radii either 2 or 4. You must place the circles side-by-side so that they only touch at one point. If exactly two of the smaller circles are used, in how many different ways may the circles be placed next to one another so that the centers of the circles are all on the same straight line and so that the sum of the lengths of the diameters of the circles is 32?

A. four
B. six
C. eight
D. ten
E. twelve
[Reveal] Spoiler: OA

Last edited by Bunuel on 11 Jan 2013, 03:10, edited 1 time in total.
Edited the question.
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Re: You are given an unlimited number of circles each of which [#permalink]  11 Jan 2013, 01:38
MOKSH wrote:
You are given an unlimited number of circles each of which having radii either 2 or 4 .you must place the circles side-by-side so that they only touch at one point. If exactly two of the smaller circles are used, in how many different ways may the circles be placed next to one another so that the centers of the circles are all on the same straight line and so that the sum of the lengths of the diameters of the circles is 32?
four
six
eight
ten
twelve

We need to have following combination
1) 2 circles of radii 2
2) 3 circle of radii 4
Now...arrangements of N things in which p and q items are similar to each other.
5! / (3!*2!)
10
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Re: You are given an unlimited number of circles each of which [#permalink]  11 Jan 2013, 03:18
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Expert's post
MOKSH wrote:
You are given an unlimited number of circles each of which having radii either 2 or 4. You must place the circles side-by-side so that they only touch at one point. If exactly two of the smaller circles are used, in how many different ways may the circles be placed next to one another so that the centers of the circles are all on the same straight line and so that the sum of the lengths of the diameters of the circles is 32?

A. four
B. six
C. eight
D. ten
E. twelve

The the sum of the lengths of the diameters of two smaller circles is 2*4=8.
Hence, the sum sum of the lengths of the diameters of larger circles is 32-8=24, which means that there should be 24/8=3 large circles.

So, we have that there should be 2 small circles and 3 large circles.

The number of arrangement is 5!/(3!*2!)=10 (the number of arrangements of 5 objects in a row, where 3 of the objects are identical and the remaining 2 objects are identical as well).

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Re: You are given an unlimited number of circles each of which [#permalink]  11 Jan 2013, 04:23
MOKSH wrote:
You are given an unlimited number of circles each of which having radii either 2 or 4. You must place the circles side-by-side so that they only touch at one point. If exactly two of the smaller circles are used, in how many different ways may the circles be placed next to one another so that the centers of the circles are all on the same straight line and so that the sum of the lengths of the diameters of the circles is 32?

A. four
B. six
C. eight
D. ten
E. twelve

Got me a little confused a bit with this statement: "If exactly two of the smaller circles are used" because I'm not a native speaker but then yeah 2 smaller circles and the rest are bigger circles....

2 smaller circles will have sum of length of diameter = 8

32-8 = 24 remaining diameter for the larger circles of diameter 8, Thus, 3 larger circles.

s s L L L

\frac{5!}{3!2!} = 10

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Re: You are given an unlimited number of circles each of which [#permalink]  11 Jan 2013, 06:43
Large circle has a diameter of 8, Small circle has a diameter of 4. Since total length of diameters must be 32 and exactly 2 smaller circles are used ---> There are 3 large circles and 2 small circles (for a total of 5 circles) that need to be arraged.
Therefore.. (5!)/(3!)(2!) = 10 (D)
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Arrangement of circles question. [#permalink]  21 Jun 2013, 06:33
You are given an unlimited number of circles each of which is identical to one of the two circles shown above. The radii of the larger circles are 4 and the radii of the smaller circles are 2. You must place the circles side-by-side so that they only touch at one point. If exactly two of the smaller circles are used, in how many different ways may the circles be placed next to one another so that the centers of the circles are all on the same straight line and so that the sum of the lengths of the diameters of the circles is 32?

A. four
B. six
C. eight
D. ten
E. Twelve
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Last edited by ankurgupta03 on 21 Jun 2013, 06:40, edited 1 time in total.
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Re: Arrangement of circles question. [#permalink]  21 Jun 2013, 06:38
Expert's post
ankurgupta03 wrote:
You are given an unlimited number of circles each of which is identical to one of the two circles shown above. The radii of the larger circles are 4 and the radii of the smaller circles are 2. You must place the circles side-by-side so that they only touch at one point. If exactly two of the smaller circles are used, in how many different ways may the circles be placed next to one another so that the centers of the circles are all on the same straight line and so that the sum of the lengths of the diameters of the circles is 32?

A. four
B. six
C. eight
D. ten
E. Twelve

I just don't get it, can some one please help. I cannot see any one of the given options to be correct.
Will post my reasoning once i receive a method to derive the answer

Please refer to the solutions above.

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Re: Arrangement of circles question. [#permalink]  21 Jun 2013, 06:39
Bunuel wrote:
ankurgupta03 wrote:
You are given an unlimited number of circles each of which is identical to one of the two circles shown above. The radii of the larger circles are 4 and the radii of the smaller circles are 2. You must place the circles side-by-side so that they only touch at one point. If exactly two of the smaller circles are used, in how many different ways may the circles be placed next to one another so that the centers of the circles are all on the same straight line and so that the sum of the lengths of the diameters of the circles is 32?

A. four
B. six
C. eight
D. ten
E. Twelve

I just don't get it, can some one please help. I cannot see any one of the given options to be correct.
Will post my reasoning once i receive a method to derive the answer

Please refer to the solutions above.

Sorry for that Bunnel, i tried searching the question, but somehow did not find it
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Math Expert
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Posts: 15211
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Kudos [?]: 15825 [0], given: 1572

Re: Arrangement of circles question. [#permalink]  21 Jun 2013, 06:42
Expert's post
ankurgupta03 wrote:
Bunuel wrote:
ankurgupta03 wrote:
You are given an unlimited number of circles each of which is identical to one of the two circles shown above. The radii of the larger circles are 4 and the radii of the smaller circles are 2. You must place the circles side-by-side so that they only touch at one point. If exactly two of the smaller circles are used, in how many different ways may the circles be placed next to one another so that the centers of the circles are all on the same straight line and so that the sum of the lengths of the diameters of the circles is 32?

A. four
B. six
C. eight
D. ten
E. Twelve

I just don't get it, can some one please help. I cannot see any one of the given options to be correct.
Will post my reasoning once i receive a method to derive the answer

Please refer to the solutions above.

Sorry for that Bunnel, i tried searching the question, but somehow did not find it

No worries. Just please name a topic properly when posting: the name of the topic must be the first sentence of the question or a string of words exactly as they show up in the question.

Thank you!
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Re: Arrangement of circles question.   [#permalink] 21 Jun 2013, 06:42
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