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A square with 3 points on each side used as vertices for triangulation (in addition to the vertices of the square) and 20 points inside the square is triangulated. What are all the possibilities for how many triangles are used in this triangulation?

Any hints or solutions will be appreciated.

I've tried looking at smaller cases, and I was looking at if it might be possible that there's only one possibility for the number of triangles. Then, how the triangles on the edges of the square can only be connected to one other point on the square, meaning it's connected to at least one point inside the square, but so far I haven't been able to get very far.

CN2020
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  • Hi CN. What have you tried? At the moment your post shows no evidence of effort. You might take a look at this, since you are new here: https://math.meta.stackexchange.com/q/9959 – 311411 May 14 '21 at 01:04
  • Hi, thank you for commenting. I've taken a look and edited the question, though I haven't been very successful so far. – CN2020 May 14 '21 at 01:11
  • Hint: There are $(20)$ internal points and $(16)$ edge points. Normally, you would presume that each collection of 3 distinct points (i.e. $\displaystyle \binom{36}{3}$ collections) generates a separate triangle. Here, however, you have to deduct the number of ways of selecting 3 distinct points such that all 3 are collinear. This will be $\displaystyle 4 \times \binom{5}{3}$. This is assuming that besides the deducted groups of 3 points, the 20 internal points are set so that none of the other collections of 3 points are collinear. – user2661923 May 14 '21 at 01:40
  • "A square with 3 points on each side used as vertices for triangulation (in addition to the vertices of the square) and 20 points inside the square": Unclear if my previous comment is interpreting this as intended. If the intent is that only those triangles should be counted where at least 2 of the vertices are from the 20 edge points, then then analysis will be similar but with a different computation. Also, if the intent is that only those triangles should be counted where at least 1 vertex is from the 20 edge points, then again, the analysis will be similar but with a diff computation. – user2661923 May 14 '21 at 01:46

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