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Fifteen dots are evenly spaced on the circumference of a circle. How

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Fifteen dots are evenly spaced on the circumference of a circle. How [#permalink]

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Fifteen dots are evenly spaced on the circumference of a circle. How many combinations of three dots can we pick from these 15 that do not form an equilateral triangle?

A. 160
B. 450
C. 910
D. 1360
E. 2640


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[Reveal] Spoiler: OA

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Fifteen dots are evenly spaced on the circumference of a circle. How [#permalink]

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New post 07 Mar 2015, 20:46
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Bunuel wrote:
Fifteen dots are evenly spaced on the circumference of a circle. How many combinations of three dots can we pick from these 15 that do not form an equilateral triangle?

A. 160
B. 450
C. 910
D. 1360
E. 2640


Kudos for a correct solution.


hi all,
there are two ways we can do this...

1) estimation/elimination process...


total ways=15C3=15!/12!3!=455...
so if total ways are 455 we eliminate any choice which is above this... so C,D and E are out...
pure logic - equilateral triangles are going to be way less than total triangles possible.... and choice A gives % of equilateral triangle almost 65% of total triangles...
so A can be eliminated ..only B left

2) pure mathematical way...


mark these points 1 to 15...
if all 15 points are equidistant, for an equilateral triangle the three points should be equidistant from each other and that will be possible only in one scenario.. when the dots are 5 portion away from each other...
so dots will be 1,6,11 or 2,7,12... and so on ..
15 points will give us only 5 sets of these values, as rest 10 will be repetition of these 5...
let me write down the five possible way..
1) 1,6,11
2) 2,7,12
3) 3,8,13
4) 4,9,14
5) 5,10,15
6) 6,11,1.. this is same as (1)..

ans 5 ways
lef ways =455-5=450 ans B....
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Re: Fifteen dots are evenly spaced on the circumference of a circle. How [#permalink]

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New post 08 Mar 2015, 15:40
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Bunuel wrote:
Fifteen dots are evenly spaced on the circumference of a circle. How many combinations of three dots can we pick from these 15 that do not form an equilateral triangle?

A. 160
B. 450
C. 910
D. 1360
E. 2640


Kudos for a correct solution.


MAGOOSH OFFICIAL SOLUTION:

Well, first of all, ignoring the type of triangle formed, how many combinations total? The easiest way to think about this is to use the Fundamental Counting Principle. For the first dot, 15 choices, then 14 left for the second choice, then 13 left for the third choice: that’s 15*14*13. But, that will count repeats: the same three dots could be chosen in any of their 3! = 6 orders, so we have to divide that number by 6. (NOTICE the non-calculator math here).
(15*14*13)/6

Cancel the factor of 3 in 15 and 6
(5*14*13)/2

Cancel the factor of 2 in the 14 and 2
(5*7*13) = 5*91 = 455

That’s how many total triangles we could create.

Of these, how many are equilateral triangles? Well, the only equilateral triangles would be three points equally spaced across the whole circle. Suppose the points are numbers from 1 to 15. From point 1 to point 6 is one-third of the circle — again, from point 6 to point 11, and from point 11 back to point 1. That’s one equilateral triangle. We could make an equilateral triangle using points
{1, 6, 11}
{2, 7, 12}
{3, 8, 13}
(4, 9, 14)
{5, 10, 15}

After that, we would start to repeat. There are five possible equilateral triangles, so 455 – 5 = 450 of these triangles are not equilateral.

Answer = (B)
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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

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Re: Fifteen dots are evenly spaced on the circumference of a circle. How [#permalink]

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New post 22 Jan 2016, 02:32
Hi Bunuel,
Please, can you kindly help me to understand why the answer is B and not D. When I got 455 (i.e., (15*14*13/3*2*1)), I read it (i.e. 455) to be the number of slots for the combinations of three-dot equilateral triangles. So I went further to multiply the 455 by 3 to get 1365 from which I deducted 5 (i.e., the 5 possible equilateral triangles) to obtain 1360, answer (D).

Thank you

Solomon
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Re: Fifteen dots are evenly spaced on the circumference of a circle. How [#permalink]

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duahsolo wrote:
Hi Bunuel,
Please, can you kindly help me to understand why the answer is B and not D. When I got 455 (i.e., (15*14*13/3*2*1)), I read it (i.e. 455) to be the number of slots for the combinations of three-dot equilateral triangles. So I went further to multiply the 455 by 3 to get 1365 from which I deducted 5 (i.e., the 5 possible equilateral triangles) to obtain 1360, answer (D).

Thank you

Solomon


Hi,
any combination of three points will give you only one triangle..
if 1,2,3 are three such points, 123, 231,312 all are same triangle..
therefore when you have got 455 ways of choosing 3 triangle, these will give you 455 triangles as each way will give you exactly one unique triangle..

Hope it helped
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Re: Fifteen dots are evenly spaced on the circumference of a circle. How [#permalink]

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New post 24 Apr 2016, 23:52
Trying to visualize it I solved it this way:

Equilateral triangle is:

XOOOOXOOOOXOOOO

So we have 15 elements with 12 and 3 repeating:
Hence C(3 and 12 identical out of 15) = C(3,15)=15!/(12!*3!)=455

No the only time we can get the equilateral triangle is as I showed earlier:

XOOOOXOOOOXOOOO
Looking at one side only:

XOOOO

There are only 5 ways the items can be arranged (i.e. XOOOO, OXOOO, OOXOO, OOOXO, OOOOX)

So the total number of triangles should be reduced by this number
and answer becomes 455-5=450.

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Re: Fifteen dots are evenly spaced on the circumference of a circle. How   [#permalink] 09 Aug 2017, 19:56
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