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# A regular polygon has 20 sides. How many triangles can be drawn by using the vertices but not using the sides?

Last updated date: 16th Jul 2024
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Hint: Here we will use the permutation and combination method to find the total number of the triangle. The total number of vertices in regular polygons is 20 and a triangle has 3 vertices. So, using permutation and combination, the total number of the triangle can be made by 20 side polygon is ${}^{20}{{C}_{3}}$. Now, we will have to remove the unwanted triangle which is made of sides. That is to remove triangles made by using one side of polygon and triangles which are made by using two sides of the polygon.

So, using permutation and combination method, the total number of triangles can be drawn is ${}^{20}{{C}_{3}}$ which we can solve by the formula ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$.
\begin{align} & {}^{20}{{C}_{3}}=\dfrac{20!}{3!17!} \\ & =\dfrac{20\times 19\times 18\times 17!}{3\times 2\times 17!} \\ & =\dfrac{20\times 19\times 18}{6} \\ & =20\times 19\times 3 \\ & =1140 \\ \end{align}
Triangles can be drawn using one side of the polygon. If we take one side, then there are a total of 16 triangles that can be drawn and we have 20 sides. So, total $20\times 16=320$ triangles.
\begin{align} & =1140-320-20 \\ & =1140-340 \\ & =800\ \text{triangles} \\ \end{align}