Today we review some more coding problems. Let us take a look at uni-color triangles. You are given  10 points in a plane such that no three of them are collinear Each pair is connected by a green dash or a blue solid line How many uni color triangles with vertices at the given points are possible.
Since we know that it takes three points to form a triangle, the total number of triangles possible is 10 choose 3 = 120. This number three is helpful. In a multicolored triangle, two edges of the triangle will be the same color. Moreover, multicolored triangles + unicolor triangles = 120.  Therefore we attempt to find unicolor triangles by finding the number of multicolored triangles. We make a table of the number of green and blue edges incident on each vertex. For e.g.:
Point   1 2 3 4 5 6 7 8 9 10
Blue    3 2 5 6 5 3 3 2 7 4
Green  6 7 4 3 4 6 6 7 2 5
We can now multiply the number of green and blue edges incident to the given points, and sum the products to find all possible multicolored triangles but each multi color triangle is counted twice because the ways to choose two edges of the same or different color still results in the same triangle. Therefore the number of uni-color triangle possible is 120 - 174/2 = 33 for the given table.
The problem is reduced from nChoose2 with complexity of O(n3) to counting n choose two edges O(n^2)  for each of the vertices.
Thus instead of writing a combinatorial coding solution as

Void CombineToTriangles (Edges a, EdgeBuilder b, int start, int level) 
{ 

   For (int I  = start ; I < a.Length; i++) 
   {  
       If (b.Length == a.Length) print b; 
       b[level] = a[i]; 
       Print b.ToTriangles(); 
       If (I < a.Length) 
            CombineToTriangles(a,b, start+1,level+1) 
       b[level] = ‘/0’; 
   } 
} 
we can reduce the problem to n!/(n-3)!3! - Sum-for-i-from-1-to-n(Blue-i * green-i / 2)

#codingexercise
Double  GetNthRootProductAlternateTermRaisedPPlusQoverPDividedQ (Double [] A,Double  p, Double q)
{
If ( A== null) return 0;
Return A.NthRootProductAlternateTermRaisedPPlusQoverPDividedQ(p, q);

}