Question 1178815


This might be a hard question. We first characterize the polygons according to the number of sides and then calculate the maximum count of them. So we will be sure that there is nothing “regular” about the distribution of the {{{5}}} points.

3-sides: There are {{{5C3=10 }}} ways to choose  {{{3}}}  of the points

4-sides: There are  {{{5C4=5}}}  ways to choose  {{{4}}}  of the points. 

For each choice, there is one concave way to join them and two self-crossing ways.  

{{{5*3=15}}}

A convenient way to think about this is to count the number of Hamiltonian Cycles in a 4-graph and that is  {{{3!/2=3}}}.

{{{5}}}-sides: The number of Hamiltonian cycles on {{{5}}} vertices is  {{{4!/2=6}}}. 

Total:  {{{10+15+6=31}}}