You are given an unlimited number of circles each of which

Question

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

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

Bunuel · Accepted Answer

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.

BangOn · Answer

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

mbaiseasy · Answer

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

sambam · Answer

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)

ankurgupta03 · Answer

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

Bunuel · Answer

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!

EMPOWERgmatRichC · Answer

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:  D

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

sheldoncooper · Answer

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

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

You are given an unlimited number of circles each of which

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

You are given an unlimited number of circles each of which

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards