Question

Find number of triangles formed in a decagon.

Answer 1

Hint: To find number of triangle in a decagon we first find number of vertices a decagon have and then using these number of vertices and number of vertices required to form a triangle in formula of combination which on simplification gives required number of triangles formed or required solution of the given problem.
Formulas used: $ ^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}} $

Complete step-by-step answer:
We know that decagon is a polygon with $ 10 $ sides. Hence, in decagon there are $ 10 $ vertices or we can say $ 10 $ point.
Also, we know that for a triangle it requires three vertices at a time.
Therefore, from our $ 10 $ given points we have to choose $ 3 $ out of them.
This can be done by using combination. As, there are $ 10 $ points available but we are required only $ 3 $ at a time.
So, number of triangle can be calculated from given $ 10 $ points by using combination are given as: $ ^{10}{C_3} $
Simplifying it using formula of combination we have $ \dfrac{{10!}}{{3!7!}} $
Or
 $
  \dfrac{{10 \times 9 \times 8 \times 7!}}{{3!7!}} \\
   \Rightarrow \dfrac{{10 \times 9 \times 8}}{{3 \times 2 \times 1}} \\
   \Rightarrow 5 \times 3 \times 8 \\
   = 120 \;
  $
Therefore, the total number of triangles that can be formed in a decagon are $ 120 $ .

Note: In decagon as there are no collinear points or vertices. Hence, from all these points we can choose any three at a time to form a triangle. Therefore, we can use the concept of combination to find the number of triangles taking n as $ 10\,\,and\,\,r\,\,as\,\,3 $ in formula of combination, which on expansion gives the total number of triangles formed in a decagon.