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You are given an unlimited number of circles each of which

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MOKSH
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You are given an unlimited number of circles each of which [#permalink] New post   Updated on: 11 Jan 2013, 04:10
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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
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D

Originally posted by MOKSH on 11 Jan 2013, 01:20.
Last edited by Bunuel on 11 Jan 2013, 04:10, 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] New post   11 Jan 2013, 04:18
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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).

Answer: D.
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Re: You are given an unlimited number of circles each of which [#permalink] New post   11 Jan 2013, 02:38
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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] New post   11 Jan 2013, 05:23
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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

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

Answer: ten
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Re: You are given an unlimited number of circles each of which [#permalink] New post   11 Jan 2013, 07:43
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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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Re: Arrangement of circles question. [#permalink] New post   21 Jun 2013, 07:39
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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.

Also, please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Pay attention to the rule #3. Thank you.!


Sorry for that Bunnel, i tried searching the question, but somehow did not find it :(
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Re: Arrangement of circles question. [#permalink] New post   21 Jun 2013, 07:42
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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.

Also, please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Pay attention to the rule #3. Thank you.!


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: You are given an unlimited number of circles each of which [#permalink] New post   30 Jan 2016, 11:04
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Hi All,

Even if you don't know the 'technical math' behind this question, you can still get to the correct answer with a little bit of 'brute force' work.

From the prompt, you can deduce that we'll need 2 small circles and 3 large circles to complete the task. From the answer choices, we know that there can only be 4, 6, 8, 10 or 12 ways to arrange those 5 circles, so it shouldn't be that tough to 'map out' the possibilities.

S = small circle
L = large circle

SSLLL
SLSLL
SLLSL
SLLLS

LSSLL
LSLSL
LSLLS

LLSSL
LLSLS

LLLSS

Total = 10 options

Final Answer:
Show Spoiler
D


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Re: You are given an unlimited number of circles each of which [#permalink] New post   09 Jul 2020, 12:15
Hi

I am unable to understand why after knowing that 24 units are left, we didn't consider any more smaller circles being placed here. The responses above have all followed - 24/8(big circle diameter) = 3 big and used combination.
It is not mentioned we cannot use more SMALL circles. Hence, shouldn't we consider the combination of 24/4 = 6 SMALL Circles ?

Thanks
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Re: You are given an unlimited number of circles each of which [#permalink] New post   06 Nov 2023, 11:09
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Re: You are given an unlimited number of circles each of which [#permalink]
06 Nov 2023, 11:09
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