BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

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

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
Moderator
Posts: 7187
Joined: Thu Sep 07, 2017 4:43 pm
Followed by:23 members

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

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

Timer

00:00

Your Answer

A

B

C

D

E

Global Stats

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



OA A

Source: Magoosh
Top
Source: — Problem Solving |
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
Top
Post new topic Post Reply
• Page 1 of 1