Select any one of the n points to be the vertex where the two congruent sides intersect. There are n ways to do that.

What you do next depends upon whether n is even or odd. If n is odd, there are now (n-1)/2 points to the "left" of the chosen vertex, (n-1)/2 on the "right". Choose any one of the (n-1)/2 points on the "left" (there are (n-1)/2 ways to do that) and the point on the "right" that completes the isosceles triangle if automatically selected also.

If n is even, there are n/2 points on either side so there are n/2 ways, rather than (n- 1)/2, to choose the second vertex.

Very similar. There are 2n+1 ways to select the first vertex, leaving (2n+1-1)/2= n points on either side. Choosing any of the n points on the "right" and any of the n points on the left guarentees that the center of the polygon will lie within the triangle.2.Find number of ways to select 3 vertices from a polygon of sides 2n+1 such that the centre of the polygon lies inside the triangle.

3.An operation on a set is said to be binary,if ,for all , and it is said to be commutative if for all .Now if then find the following

(i)Total number of binary operations on

(ii)Total number of binary operations on such that

(iii)Total number of binary operations on such that