Creating Triangles: How Many Can You Make?

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Discussion Overview

The discussion revolves around calculating the number of triangles that can be formed using a specific arrangement of dots, with a focus on combinatorial methods. The context includes mathematical reasoning and combinatorial counting techniques.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant outlines three methods for forming triangles based on the arrangement of dots, specifying the combinations of dots from horizontal and vertical locations.
  • The first method involves selecting one dot from the vertical column and one from the horizontal row, leading to a calculation of 7C1 x 4C1.
  • The second method involves selecting two dots from the horizontal row and one from the vertical column, calculated as 7C2 x 4C1.
  • The third method involves selecting one dot from the horizontal row and two from the vertical column, calculated as 7C1 x 4C2.
  • Another participant confirms the calculations for each method and sums them to arrive at a total of 154 triangles.

Areas of Agreement / Disagreement

Participants appear to agree on the methods and calculations presented for counting the triangles, with no evident disagreement on the results.

Contextual Notes

The discussion relies on combinatorial principles, and the assumptions regarding the arrangement of dots and the definitions of combinations are not explicitly stated.

MichaelLiu
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How many triangles can be made given the following dots?

Triangle.jpg


Thanks a lot!
 
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Apart from the location at the top left corner of the diagram, there are seven locations on the horizontal row, and four locations on the vertical column. There are three ways to place dots on three of these locations to form a triangle:

First way: one dot at the corner, one from the row of seven, and one from the column of four;
Second way: two dots on the row of seven, and one on the column of four (as in the above diagram);
Third way: one dot on the row of seven, and two on the column of four.

Can you count the number of triangles formed in each of these three ways?
 
Hey @Opalg ! Thank you for your response!

Would this be the correct answer?

First way:
7C1 x 4C1 = 28

Second way:
7C2 x 4C1 = 84

Third way:
7C1 x 4C2 = 42

28 + 84 + 42 = 154 triangles
 
MichaelLiu said:
Hey @Opalg ! Thank you for your response!

Would this be the correct answer?

First way:
7C1 x 4C1 = 28

Second way:
7C2 x 4C1 = 84

Third way:
7C1 x 4C2 = 42

28 + 84 + 42 = 154 triangles
Looks good to me. (Yes)
 

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