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In the above diagram, the 16 dots are in rows and columns, [#permalink]
03 Feb 2014, 10:47

Expert's post

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00:00

A

B

C

D

E

Difficulty:

55% (hard)

Question Stats:

33% (02:10) correct
67% (01:55) wrong based on 49 sessions

Attachment:

4x4 grid.JPG [ 9.78 KiB | Viewed 790 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/

Re: In the above diagram, the 16 dots are in rows and columns, [#permalink]
03 Feb 2014, 12:04

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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. _________________

Paras.

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Re: In the above diagram, the 16 dots are in rows and columns, [#permalink]
03 Feb 2014, 23:56

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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

Re: In the above diagram, the 16 dots are in rows and columns, [#permalink]
30 Mar 2014, 08:04

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 ?

Re: In the above diagram, the 16 dots are in rows and columns, [#permalink]
30 Mar 2014, 09:13

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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 523 times ]

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