Solution:
Now, let us assume that one vertex of the required triangle as
.Now, let us find the number of triangles by considering the number of pairs of remaining sides of the triangle.
Let us assume that the remaining vertices of the triangle as
where, represents the vertex number.To find the number of triangles that contain the center of the polygon.
We know that for a
sided polygon the gap between the other two vertices must be at most 
, that is the gap should be or less than in order to get the center of the polygon in the triangle.Here, we considered the first vertex as
.By using the above condition the possible vertices of
are 
and .Now, let us take the possibilities of b for each vertex of
. (1) For
the possible vertex of 
is 
.(2) For
the possible vertex of is or 
.(3) For
the possible vertex of b is or or 
.(4) For
the possible vertex of b is or or or 
.Here, we can see that there are total of
triangles that contain the center of the polygon in the triangle.Let us assume that the number of total number of outcomes as
then we get,
Let us assume that the total number of outcomes as 
.Here, we can see that we have already selected one vertex as
then the number of ways of selecting the remaining 
vertices from vertices is given as
We know that the formula of probability aBy using the formula we get the required probability as
We are given that the required probability as 
.By comparing the two probabilities we get
.Hence, Option (1) is the correct answer.
