The regular 3 by 3 grid of dots above consists of evenly spaced rows

Question

https://gmatclub.com/forum/download/file.php?id=55799 The regular 3 by 3 grid of dots above consists of evenly spaced rows and columns, in which the rows are perpendicular to the columns. How many different ways can a line segment be drawn from one dot in the grid to another, without intersecting any other dot? A. 22 B. 23 C. 24 D. 26 E. 28 1.jpg

HouseStark · Accepted Answer

The no. of lines can be formed using 9 dots are,

as to form a single line you require 2 dots therefore= =  9C2

this includes line containing both 2 dots line and 3 dots line, eliminating those 3 dots line we get line segment one dot in the grid to another, without intersecting any other dot

there are total 8 such line segments intersecting 1 extra dot those are

AB, BC, CD, DA, EG, FH, AC, BD=8 LINE SEGMENTS

THUS,  9C2-8=36-8=28

ANSWER E

nkme2007 · Answer

5 lines from each dot at the corner.

4 lines from the centre dot excluding the lines to corners.

4 lines from centre to side dots

Total: (5*4)+4+4 = 28

unraveled · Answer

Let the rows be 1, 2 and 3 and columns be A, B and C. Thus the dots are A1, A2, A3, B1, B2, B3, C1, C2 and C3. Total line segments = line segments having 3 dots + line segments having only 2 dots. Since line segments can have 2 dots, total line segments = 9C2 = 36 (including line segments with 3 dots) Now, we can count line segments with 3 dots also. There are 8 of them as follows: A1A2A3 B1B2B3 C1C2C3 A1B1C1 A2B2C2 A3B3C3 A1B2C3 A3B2C1 Total line segments with 2 dots = 36 - 8 = 28. Note: The line segments with 3 dots can be counted as two line segments with 2 dots. Visually, required possibilities are: - Horizontal: A1B1, B1C1, A2B2, B2C2, A3B3, B3C3 - Vertical: A1A2, A2A3, B1B2, B2B3, C1C2, C2C3 - 45° downhill: A1B2, B2C3, B1C2, A2B3 - <45° downhill: A1C2, A2C3 - >45° downhill: A1B3, B1C3 - 45° uphill: A3B2, B2C1, A2B1, B3C2 - <45° uphill: A2C1, A3C2 - >45° uphill: A3B1, B3C1 Answer E.

NitishJain · Answer

IMO  E

I tried to count like in topLHS square: 6 lines possible
top RHS :5
bottom LHS: 5
bottom RHS: 4
then  two  lines in top as diagonals and  two  in bottom
then  two  lines from top extreme dots to center dot bottom
and similarly  two  lines from bottom extreme dots to center top.

Total: 28

Nannicka · Answer

Total is 24, there are 5 lines from the corners, times 4 is 24. Then there are 4 lines from the middle dot to the sides. So that is 24. If you take all lines from the centre you would get 28 but that counts the lines between middle and corners double

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ouslam · Answer

the only solution for these sort of problems is to count all possible cases and guess (I guessed the answer E after counting 27 unique value with this method )

first I took 4 dots forming a square(bottom left) to form a line 4C2 = 6 lines , you also have a square on top right (+6)

then you have (from the top left point and bottorm right point ) 3 *2 new distinct lines 
also middle point right with bottom center, and the parallel line to this line (according to the center of gravity of the whole square) +2

you have also 4 rectangles (top left point , middle top point , bottom left point , bottom middle point) you have 8

6+6+3*2+2+8=28 , E

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The regular 3 by 3 grid of dots above consists of evenly spaced rows

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