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# What is the greatest possible number of points at which 11 circles wit

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Math Expert
Joined: 02 Sep 2009
Posts: 65184
What is the greatest possible number of points at which 11 circles wit  [#permalink]

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19 Dec 2015, 07:43
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Difficulty:

45% (medium)

Question Stats:

68% (01:48) correct 32% (01:51) wrong based on 267 sessions

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Math Revolution and GMAT Club Contest Starts!

QUESTION #14:

What is the greatest possible number of points at which 11 circles with different radii intersected one another?

A. 45
B. 60
C. 85
D. 90
E. 110

Check conditions below:

Math Revolution and GMAT Club Contest

The Contest Starts November 28th in Quant Forum

We are happy to announce a Math Revolution and GMAT Club Contest

For the following four (!) weekends we'll be publishing 4 FRESH math questions per weekend (2 on Saturday and 2 on Sunday).

To participate, you will have to reply with your best answer/solution to the new questions that will be posted on Saturday and Sunday at 9 AM Pacific.
Then a week later, the forum moderator will be selecting 2 winners who provided most correct answers to the questions, along with best solutions. Those winners will get 6-months access to GMAT Club Tests.

PLUS! Based on the answers and solutions for all the questions published during the project ONE user will be awarded with ONE Grand prize:

PS + DS course with 502 videos that is worth $299! All announcements and winnings are final and no whining GMAT Club reserves the rights to modify the terms of this offer at any time. NOTE: Test Prep Experts and Tutors are asked not to participate. We would like to have the members maximize their learning and problem solving process. Thank you! _________________ ##### Most Helpful Community Reply Intern Joined: 21 Jan 2013 Posts: 36 Concentration: General Management, Leadership Schools: LBS GPA: 3.82 WE: Engineering (Computer Software) Re: What is the greatest possible number of points at which 11 circles wit [#permalink] ### Show Tags 20 Dec 2015, 03:30 8 3 Any Circle intersects another circle at maximum two intersection points. With two circles : greatest number of intersection points = 2 With third circle introduced to the first two (already intersecting at two points), the third circle can intersect both the circles at two different intersection points, introducing 4 more intersection points ( third circle cuts each of the first two at two different poitns) . in addition to the two intersection points of first two. i.e., with three circles , max # of intersection points = 2 + 2*2 = 6 with 4th circle introduced, it cuts the first three circles at two intersection points each i.e, 2*3 additional intersection points introduced now. with 4 circles , max # of intersection points = (2 + 2*2) (intersection points for 3 circles) + 2*3 (additional intersection points introduced by 4th circle) = 2 + 2*2 + 2*3 Similarily , with 5th circle . total = 2 + 2*2 + 2*3 + 2*4 we see a series emerging here greatest number of of intersection points for m circles = 2 + 2*2 + 2*3 + .....+ 2*(m-1) Applying the series extrapolation for 11 circles, greatest number of intersection points = 2 + 2*2 + 2*3 + .....+ 2*10 = 2 * (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) = 2 * (110)/2 = 110 ( calculation for the sum shown below) -- Calculation for the sum. since the sum in the paranthesis is a sum of consecutive integers , we can calculate the average by (last + first) / 2 Sum in paranthesis = average * # of terms = 10*((1 + 10)/2) = 110/2 -- Correct Answer : E ##### General Discussion Intern Joined: 12 Jul 2015 Posts: 17 Location: United States Concentration: Strategy, General Management WE: General Management (Pharmaceuticals and Biotech) Re: What is the greatest possible number of points at which 11 circles wit [#permalink] ### Show Tags 19 Dec 2015, 11:05 5 3 The first 2 circles intersect at 2 points. 3rd circle can intersect the first two circles at 4 points. refer to the figure. i.e Circle 1 = no intersection Circle 2 = 2 points Circle 3 = 2+4 points Circle 4 = 2+4+8 points . . . Circle 11= 2+4+8+10+12+14+16+18+20 = n(n+1)= 10*11=110 points. Implies N circles can be drawn to intersect N(N-1) points . In this case, 11 circles can be drawn that can intersect at 11*10=110 points Ans E Attachments File comment: circles circles.png [ 27.71 KiB | Viewed 6215 times ] Marshall & McDonough Moderator Joined: 13 Apr 2015 Posts: 1665 Location: India Re: What is the greatest possible number of points at which 11 circles wit [#permalink] ### Show Tags 21 Dec 2015, 19:43 5 2 Given: Number of circles with different radii = 11 We can draw a few circles to obtain the below results For 2 circles, max number of points of intersection = 2 = 2 * 1 For 3 circles, max number of points of intersection = 6 = 3 * 2 For 4 circles, max number of points of intersection = 12 = 4 * 3 We can observe a pattern here. For n circles with different radii, max number of points of intersection can be deduced from the above results as n * (n-1). Therefore, for 11 circles with different radii, max number of points of intersection = 11 * 10 = 110 Answer: E Intern Joined: 30 Sep 2013 Posts: 20 Re: What is the greatest possible number of points at which 11 circles wit [#permalink] ### Show Tags 19 Dec 2015, 09:43 3 3 Answer is E i.e 110 Two circles can intersect at maximum 2 points. Three circles can intersect at maximum 6 points.. Because we select 2 circles from 3 and each two circles can intersect at maximum 2 points 2$$c^3_2$$ ... Three circles can intersect at 6 points So 11 circles can intersect at maximum $$2c^{11}_2$$ ... 2 * 11 * 5 = 110 Intern Joined: 01 Nov 2015 Posts: 29 Location: India Concentration: Marketing, Entrepreneurship WE: Engineering (Computer Software) Re: What is the greatest possible number of points at which 11 circles wit [#permalink] ### Show Tags 20 Dec 2015, 19:46 2 1st Circle doesn't intersect itself - 2*(1-1)=0 points 2nd Circle intersects 1st at 2*(2-1)=2 Points 3rd Circle intersect 2nd and 3rd at 2*(3-1)=4 Points 4th Circle intersects other Circles at 2*(4-1)=6 points ... 11th circle intersects other circles at 2*(11-1)=20 points Therefore total no of points = 2 + 4 + 6 + ... + 20 $$Sum = 10/2 * (20+2) = 5*22 =110$$ Answer is E Intern Joined: 09 Jul 2015 Posts: 45 Re: What is the greatest possible number of points at which 11 circles wit [#permalink] ### Show Tags 21 Dec 2015, 16:54 2 Answer is E -> 110. 2 circle intersects at 2 points, 3 circles intersect at 4 points, 4 circles intersect at 6 points, 5 circles at 8 points, etc., Hence max # of points will be the result of addition of these points - as you can draw a circle anyway you want to ensure there are distinct points with each of these circle's intersection. S = 2+4+6+... +20 (n=10) as there are 11 circles S = (a1+an)/2*n = (2+20)/2*10 = 110 Intern Joined: 17 Aug 2014 Posts: 10 Re: What is the greatest possible number of points at which 11 circles wit [#permalink] ### Show Tags 24 Dec 2015, 09:21 2 The answer is E By drawing circles and deducing 2 circles intersect at 2 points(1x2) 3 circles intersect at 6 points(2x3) 4 circles intersect at 12 points(3x4) 11 circles intersect at 10x11=110 points Intern Joined: 05 Jun 2013 Posts: 25 Re: What is the greatest possible number of points at which 11 circles wit [#permalink] ### Show Tags 25 Dec 2015, 04:52 1 1 2 circles can intersect at the most 2 points. ---> which is 2P2 3 circles can intersect at the most 6 points. ----> which is 3P2 Therefore it is similar to number of arrangements possible for 11 objects taken two at a time. Therefore 11 circles with different radii will intersect at 11P2 = 11!/9! = 11*10= 110 points. Answer choice E. Intern Joined: 04 Sep 2015 Posts: 35 Location: Germany Concentration: Operations, Finance WE: Project Management (Aerospace and Defense) Re: What is the greatest possible number of points at which 11 circles wit [#permalink] ### Show Tags 27 Dec 2015, 02:14 1 1 1 circles have 0 intersection point ( 1 x 0) 2 circles have 2 intersection points (2 x 1) 3 circles have 6 intersection points (3 x 2) 4 circles have 12 intersection points (4 x 3) .... .... a pattern develops! So, 11 circles have (11 x 10) = 110 intersection points However, an expression exists as well ie; C x (C-1) = I, where C is the number of circles and I the number of intersections. Math Expert Joined: 02 Sep 2009 Posts: 65184 Re: What is the greatest possible number of points at which 11 circles wit [#permalink] ### Show Tags 27 Dec 2015, 11:17 Bunuel wrote: Math Revolution and GMAT Club Contest Starts! QUESTION #14: What is the greatest possible number of points at which 11 circles with different radii intersected one another? A. 45 B. 60 C. 85 D. 90 E. 110 Check conditions below: Math Revolution and GMAT Club Contest The Contest Starts November 28th in Quant Forum We are happy to announce a Math Revolution and GMAT Club Contest For the following four (!) weekends we'll be publishing 4 FRESH math questions per weekend (2 on Saturday and 2 on Sunday). To participate, you will have to reply with your best answer/solution to the new questions that will be posted on Saturday and Sunday at 9 AM Pacific. Then a week later, the forum moderator will be selecting 2 winners who provided most correct answers to the questions, along with best solutions. Those winners will get 6-months access to GMAT Club Tests. PLUS! Based on the answers and solutions for all the questions published during the project ONE user will be awarded with ONE Grand prize: PS + DS course with 502 videos that is worth$299!

All announcements and winnings are final and no whining GMAT Club reserves the rights to modify the terms of this offer at any time.

NOTE: Test Prep Experts and Tutors are asked not to participate. We would like to have the members maximize their learning and problem solving process.

Thank you!

MATH REVOLUTION OFFICIAL SOLUTION:

The greatest possible number of points of intersection between two circles is 2. The greatest possible number of points of intersection of three circles is 2+2*2 (The greatest possible number of points of intersection between two circles + The greatest possible number of points of intersection that one circle can have with other two circles). The greatest possible number of points of intersection of four circles is 2+2*2+2*3 (The greatest possible number of points of intersection of three circles + The greatest possible number of points of intersection that one circle can have with other two circles).

So, the greatest possible number of points of intersection of 11 circles is, 2+2*2+2*3+…..2*10=2(1+2+…..+10)=2[10(10+1)/2]=110. The correct answer is E.

Note: 1+2+…..+n=n(n+1)/2
_________________
Re: What is the greatest possible number of points at which 11 circles wit   [#permalink] 27 Dec 2015, 11:17