Question

the number of triangle formed by the vertices of the decagonsuch that at least one side is common​

Answer 1

For the case we consider the equation 

Total number of triangles formed 

 Triangle having no side common triangle having exactly one side common triangles having exactly two sides common with those of the polygon

 Number of triangles having no sides common with that of polygon 

Total Number of triangles i.e Number of  exactly one side common  Number of triangles having exactly two sides common.

Now Number of  having exactly one side common 

and Number of triangles having exactly two sides common.

For this we have to select three consecutive vertices of the polygon,

If vertices of polygon are marked by 

i.e