SOLUTION: how many polygons can be possibly formed from 5 distinct points on a plane,no three of which are collinera

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Question 1178815: how many polygons can be possibly formed from 5 distinct points on a plane,no three of which are collinera
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

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%2A3=15
A convenient way to think about this is to count the number of Hamiltonian Cycles in a 4-graph and that is 3%21%2F2=3.
5-sides: The number of Hamiltonian cycles on 5 vertices is 4%21%2F2=6.
Total: 10%2B15%2B6=31