How many triangles?

How many triangles can you draw using points from the following figure?

dots_for_triangles1080×548 2.47 KB

You can make 155 triangles from the figure. I’ve been trying to figure out how to insert the symbols I’m looking for but am struggling with that, so I’m going to give it my best shot with what I know how to do. To find the number of triangles I multiplied (6 choose 2) with (5 choose 1) first, because you can make a triangle using two dots from the top arc, and one dot from the bottom row. I then added the product of (6 choose 1) and (5 choose 2) because you can use one dot from the top arc, and two dots from the bottom row to make a triangle. I then added (6 choose 3), because you can make a triangle using the dots in the arc alone. If you add all of these together you get 155.

@asmith42

Can you write that as a sum or difference of binomials?

So, I guess that’s

Note that 5\choose0 is just 1 but I included it to emphasize the relationships I see with the others.

Maybe - I didn’t compute it. I think the expression with the binomials is the ideal way to write it.

Gotcha, I’ll use the expression with binomials on the quiz. What I have written on my paper does look like what you input up above though, so I feel good about that.

I think this would be (11 choose 3) - (5 choose 3) OR (6 choose 3)(5 choose 0) + (6 choose 2)(5 choose 1)+ (6 choose 1)*(5 choose 2)