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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] New post 11 Jan 2013, 00:20
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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
[Reveal] Spoiler: OA

Last edited by Bunuel on 11 Jan 2013, 03:10, edited 1 time in total.
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Re: You are given an unlimited number of circles each of which [#permalink] New post 11 Jan 2013, 01:38
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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, 03:18
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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, 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

Answer: ten
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Re: You are given an unlimited number of circles each of which [#permalink] New post 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] New post 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] New post 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.

Also, please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Pay attention to the rule #3. Thank you.!
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Re: Arrangement of circles question. [#permalink] New post 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.

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, 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.

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!
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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