Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 46333

Fifteen dots are evenly spaced on the circumference of a circle. How [#permalink]
Show Tags
06 Mar 2015, 07:34
Question Stats:
59% (01:37) correct 41% (01:25) wrong based on 118 sessions
HideShow timer Statistics



Math Expert
Joined: 02 Aug 2009
Posts: 5938

Fifteen dots are evenly spaced on the circumference of a circle. How [#permalink]
Show Tags
07 Mar 2015, 20:46
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 =4555=450 ans B....
_________________
Absolute modulus :http://gmatclub.com/forum/absolutemodulusabetterunderstanding210849.html#p1622372 Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
GMAT online Tutor



Math Expert
Joined: 02 Sep 2009
Posts: 46333

Re: Fifteen dots are evenly spaced on the circumference of a circle. How [#permalink]
Show Tags
08 Mar 2015, 15:40
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 noncalculator 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 onethird 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)
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
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.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Manager
Joined: 02 Jun 2015
Posts: 192
Location: Ghana

Re: Fifteen dots are evenly spaced on the circumference of a circle. How [#permalink]
Show Tags
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 threedot 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
_________________
Kindly press kudos if you find my post helpful



Math Expert
Joined: 02 Aug 2009
Posts: 5938

Re: Fifteen dots are evenly spaced on the circumference of a circle. How [#permalink]
Show Tags
22 Jan 2016, 03:00
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 threedot 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
_________________
Absolute modulus :http://gmatclub.com/forum/absolutemodulusabetterunderstanding210849.html#p1622372 Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
GMAT online Tutor



Manager
Joined: 16 Feb 2016
Posts: 53
Concentration: Other, Other

Re: Fifteen dots are evenly spaced on the circumference of a circle. How [#permalink]
Show Tags
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 4555=450.



NonHuman User
Joined: 09 Sep 2013
Posts: 7060

Re: Fifteen dots are evenly spaced on the circumference of a circle. How [#permalink]
Show Tags
09 Aug 2017, 19:56
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up  doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Books  GMAT Club Tests  Best Prices on GMAT Courses  GMAT Mobile App  Math Resources  Verbal Resources




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






