The official solution may actually be correct if we are counting triangles that have also been divided into smaller triangular/polygonal regions by the n lines.  Otherwise, it does not appear there is a simple expression that provides the maximum number of triangles that can be formed with n straight lines.  From The On-Line Encyclopedia of Integer Sequences (Sloan A107427), we have this small list:

n    1   2   3   4   5   6   7   8   9  10 ...   
max  0   0   1   2   4   7  10  14  18  22 ...

Yet it should be noted that this sequence of terms excludes cases where the closed areas also create polygons that are not triangular and does not count the "larger" triangles (triangles that have been subdivided into smaller triangular/polygonal regions).

Edited on March 14, 2008, 11:26 pm

Posted by Dej Mar on 2008-03-14 15:22:18