4 triangles can be created with 3 points on a line.
20 triangles can be created with 6 points on a line.
30 triangles can be created with 7 points on a line.
Combination has to do with selection.
Note that a triangle has 3 vertices. If there are 4 points on a line, the number of triangles it is possible to create is derived using the combination formula shown as:
[tex]nC_r=\frac{n!}{(n-r)!r!}[/tex]
[tex]4C_3 = \frac{4!}{(4-3)!3!}\\4C_3 =\frac{4\times3!}{3!}\\4C_3=4triangles[/tex]
This means that 4 triangles can be created with 3 points on a line.
For a line with 6 points:
[tex]6C_3 = \frac{6\times 5\times 4\times 3!}{(6-3)!3!}\\6C_3 =\frac{6\times5\times 4\times3!}{3!3!}\\4C_3=20triangles[/tex]
This shows that 20 triangles can be created with 6 points on a line.
For a line with 7 points:
[tex]7C_3 = \frac{7\times 6\times 5\times 4!}{(7-3)!3!}\\4C_3 =\frac{7\times6\times 5\times4!}{4!3!}\\4C_3=30triangles[/tex]
This shows that 30 triangles can be created with 7 points on a line.
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