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

Question

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: https://magoosh.com/gmat/2014/gmat-math- ... -formulas/ Mike :-)

ganeshrkamath · Accepted Answer

Hope this helps: 4x4 grid.JPG Cheers

soods26 · Answer

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.

ganeshrkamath · Answer

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  A

Cheers

GDR29 · Answer

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 ?

GDR29 · Answer

very nice! Thks a lot !

Fdambro294 · Answer

Given that there are 16 dots equally spaced in an array, any unique combination of E does chosen can be a valid triangle with positive area UNLESS those 3 points chosen are COLINEAR

to find the number of valid triangles:

(total possible unique combinations of 3 dots chosen out of the 16 total)

-

(Unfavorable Outcomes in which the 3 dots chosen all lie on the same line)

(16 c 3) = 16! / (3! 13!) = 560 unique combinations can be made of 3 dots

Out of these 560 groupings, how many groupings involve 3 dots located on the same Line

Case 1: vertical rows

For any one vertical row, there are 4 dots. Any 3 unique dots chosen from those 4 dots will lie on a straight line and will NOT form a valid triangle. How many ways can we have unique groups of 3 dots chosen out of 4 in a row?

(4 c 3) = 4 ways

Since there are 4 rows: (4 rows) * (4 combinations per row that do not form a triangle) = 16 invalid

Case 2: Same logic as case 1.

There will be another 16 invalid groupings.

Case 3: 3 dots that lie on Diagonally Upwards Sloping Lines

There are 2 Diagonally upward slopping lines that sandwich the Center Diagonal.

Each of those lines has only 3 dots. Each will be an invalid case ———> 2 invalid

The upward sloping diagonal of the 4 by 4 array contains 4 dots. Any 3 dots chosen from these 4 dots that lie on a diagonal straight line will NOT form a triangle.

(4 c 3) = 4 invalid

6 invalid cases

Case 4: downward sloping diagonal lines

Symmetrically, the same logic will apply as applied in case 3.

Another 6 invalid cases that must be removed.

Total number of possible triangles = (560) - (16) - (16) - (6) - (6) =

(560) - (44) =

516 possible triangles can be made

Answer (A)

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

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

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