136 triangles can be formed from a 20‑gon using diagonals from a single common vertex.

Core idea

Pick one vertex of the 20‑gon and insist that every triangle you form must include this fixed vertex.

To complete a triangle you just need to choose 2 more vertices from the remaining vertices that can connect to it by diagonals.

Counting the usable diagonals

  • A 20‑gon has 20 vertices in total.
  • From the chosen vertex, you cannot connect to:
    • itself, and
    • its two adjacent neighbors (those are sides, not diagonals).
  • So the number of vertices you can connect to by diagonals is 20−3=1720-3=1720−3=17.

Forming triangles

Each triangle that uses the fixed vertex is determined by choosing 2 of those 17 diagonal endpoints.

The number of ways to do that is the combination (172)=17⋅162=136\binom{17}{2}=\frac{17\cdot 16}{2}=136(217​)=217⋅16​=136.

Final answer

Using diagonals from a common vertex of a 20‑gon, the number of distinct triangles that can be formed is 136.

TL;DR: Fix one vertex, count its 17 diagonal endpoints, then compute (172)=136\binom{17}{2}=136(217​)=136.

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