In the above diagram, the 16 dots are in rows and columns, and are equally spaced in both the horizontal & vertical

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by BTGmoderatorDC » Wed Dec 08, 2021 1:17 am

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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

OA A

Source: Magoosh

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Source: Beat The GMAT — Problem Solving | Wed Dec 08, 2021 1:17 am

regor60: Master | Next Rank: 500 Posts; Posts: 416; Joined: Thu Oct 15, 2009 11:52 am; Thanked: 27 times

Re: In the above diagram, the 16 dots are in rows and columns, and are equally spaced in both the horizontal & vertical

by regor60 » Wed Dec 08, 2021 5:02 am

Need 3 points for a triangle, but only 2 points can be collinear. Setting that issue aside for the moment, 3 points can be selected
16!/13!3! = 560

That's the maximum number of triangles, but we know a bunch won't work because some of the groups of 3 will lie on the same line. But we can eliminate C,D,E as answers.

Now we need to identify the number of sets of 3 points that lie on a line and subtract.

Each row and column has 4 points. 3 points can be selected from each
4!/3!1! = 4. Since there are 8 rows and columns, this eliminates 32.

But we have major and minor diagonals also that are lines.

The two major diagonals have 4 points each, so by the logic above this eliminates another 8.

Each major diagonal has a minor diagonal on either side comprising 3 points each.

3 points can be selected from 3
1 way, multiplied by 4 minor diagonals eliminates another 4.

Total number of 3 point sets to be eliminated is 32+8+4=44

Total number of triangles = 560-44=516,A

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