Question

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

Answer 1

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.

Complete step-by-step answer:

We know that the regular polygon has 20 sides and a triangle has 3 sides.
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 ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$.
Applying the formula,
$\begin{align}
  & {}^{20}{{C}_{3}}=\dfrac{20!}{3!17!} \\
 & =\dfrac{20\times 19\times 18\times 17!}{3\times 2\times 17!} \\
 & =\dfrac{20\times 19\times 18}{6} \\
 & =20\times 19\times 3 \\
 & =1140 \\
\end{align}$
Now, we have to remove the unwanted triangle that is triangles can be drawn using one side of polygon and triangles can be drawn using two side of polygon.
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.
Now, there will be a total 20 triangles that can be drawn using two sides of the polygon.
So, total triangle can be drawn by vertices not using side,
$\begin{align}
  & =1140-320-20 \\
 & =1140-340 \\
 & =800\ \text{triangles} \\
\end{align}$

Note: Students may make a mistake that is they do not remove the unwanted triangle which can lead to the wrong answer i.e. 1140 triangles. Also, sometimes students will not consider both cases, i.e. triangle formed using 1 side and 2 sides. So, they might get a different answers.