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# In the above diagram, the 16 dots are in rows and columns,

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Magoosh GMAT Instructor
Joined: 28 Dec 2011
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In the above diagram, the 16 dots are in rows and columns, [#permalink]

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03 Feb 2014, 11:47
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33% (01:33) correct 67% (02:07) wrong based on 97 sessions

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4x4 grid.JPG [ 9.78 KiB | Viewed 4068 times ]

In the above diagram, the 16 dots are in rows and columns, and are equally spaced in both the horizontal & vertical direction. How many triangles, of absolutely any shape, can be created from three dots in this diagram? Different orientations (reflections, rotations, etc.) and/or positions count as different triangles. (Notice that three points all on the same line cannot form a triangle; in other words, a triangle must have some area.)
(A) 516
(B) 528
(C) 1632
(D) 3316
(E) 3344

Many GMAT math problems, such as this one, cannot be solved by formulas alone. For a discussion of the uses & abuses of formulas on the GMAT Quant section, as well as the complete solution to this problem, see:
http://magoosh.com/gmat/2014/gmat-math- ... -formulas/

Mike
[Reveal] Spoiler: OA

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Mike McGarry
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Re: In the above diagram, the 16 dots are in rows and columns, [#permalink]

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03 Feb 2014, 13:04
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Number of ways to select 3 points out of 16 is 16C3 = 560. There is possibility that in some cases out of the 560 cases, the three points lie on the same line and therefore do not form a triangle. This eliminates options B,C and D

There are 4 columns and 4 rows make it a total of 8 linear possible arrangements of the points. Number of ways in which the points can be arranged along each row or column = 8*(4C3) = 8*4 = 32.

We are left with 560-32 =528 ways. Now there is also the possibility that the three points fall on a straight line if placed along the diagonal. Thus the number of ways is definitely less than 528, leaving option A.
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Re: In the above diagram, the 16 dots are in rows and columns, [#permalink]

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04 Feb 2014, 00:56
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mikemcgarry wrote:
Attachment:
4x4 grid.JPG

In the above diagram, the 16 dots are in rows and columns, and are equally spaced in both the horizontal & vertical direction. How many triangles, of absolutely any shape, can be created from three dots in this diagram? Different orientations (reflections, rotations, etc.) and/or positions count as different triangles. (Notice that three points all on the same line cannot form a triangle; in other words, a triangle must have some area.)
(A) 516
(B) 528
(C) 1632
(D) 3316
(E) 3344

Many GMAT math problems, such as this one, cannot be solved by formulas alone. For a discussion of the uses & abuses of formulas on the GMAT Quant section, as well as the complete solution to this problem, see:
http://magoosh.com/gmat/2014/gmat-math- ... -formulas/

Mike

Number of ways to connect any 3 distinct dots = 16C3 = (16*15*14)/(3*2*1) = 560
Number of ways to connect any 3 distinct dots into a horizontal line (non-triangles) = 4C3*4 = 16
Number of ways to connect any 3 distinct dots into a vertical line (non-triangles) = 4C3 *4 = 16
Number of ways to connect any 3 distinct dots into top-left to bottom-right lines (non-triangles) = 1+4C3+1 = 6
Number of ways to connect any 3 distinct dots into bottom-left to top-right lines (non-triangles) = 1+4C3+1 = 6

Number of ways to connect any 3 distinct dots in the figure into a triangle = 560 - 16 - 16 - 6 - 6 = 516

Choose
[Reveal] Spoiler:
A

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Kudos [?]: 21 [1], given: 14

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Re: In the above diagram, the 16 dots are in rows and columns, [#permalink]

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30 Mar 2014, 09:04
Hello,
Please can someone explain how to calculate the number number of ways to connect any 3 distinct dots into top-left to bottom-right line and into bottom-left to top-right lines.

Why is not 4C3 *2 ?? Why you need to sum 1+ 4C3 +1 ?

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Manager
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Re: In the above diagram, the 16 dots are in rows and columns, [#permalink]

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30 Mar 2014, 10:13
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GDR29 wrote:
Hello,
Please can someone explain how to calculate the number number of ways to connect any 3 distinct dots into top-left to bottom-right line and into bottom-left to top-right lines.

Why is not 4C3 *2 ?? Why you need to sum 1+ 4C3 +1 ?

Hope this helps:
Attachment:

4x4 grid.JPG [ 44.21 KiB | Viewed 3770 times ]

Cheers
_________________

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Kudos [?]: 21 [6], given: 14

Intern
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Re: In the above diagram, the 16 dots are in rows and columns, [#permalink]

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30 Mar 2014, 18:46
very nice! Thks a lot !

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Re: In the above diagram, the 16 dots are in rows and columns, [#permalink]

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10 Aug 2017, 20:54
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Re: In the above diagram, the 16 dots are in rows and columns,   [#permalink] 10 Aug 2017, 20:54
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