Answer

Verified

358.1k+ views

**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.

Recently Updated Pages

Write a composition in approximately 450 500 words class 10 english JEE_Main

Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main

What is the common property of the oxides CONO and class 10 chemistry JEE_Main

What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main

If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main

The area of square inscribed in a circle of diameter class 10 maths JEE_Main

Other Pages

Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main

Electric field due to uniformly charged sphere class 12 physics JEE_Main

Excluding stoppages the speed of a bus is 54 kmph and class 11 maths JEE_Main

The correct order of penetrating power is A gamma alpha class 11 chemistry JEE_Main

If a wire of resistance R is stretched to double of class 12 physics JEE_Main