determining state space based on area

Question

I have been tasked with figuring out a state space for a problem based on the area of a rectangle. It seems that I have made my state space far too large and need some feedback.
So far I have an area that has a value fo 600 for a y axis and 300 for an x axis. I determined the number of points to be

(600 x 300) ! or 180,000!

Therefore my robot would need to inspect this many potential spaces, before I apply an algorithm.

This number seems quite high and if that is the case it would make my problem unsolveable before I die especially if I implement the algorithm incorrectly. Any help would be greatly appreciated especially if my math is off in determining the number of points.

EDIT I was under the impression to see how many pairs of points you would have to take the cartesian product of the total available points. Which in turn would be (600x300)! . If this is incorrect please let me know.

Answer 1

First of all, the number of "points" (as defined in mathematics - the only relevant definition) in a rectangle of any size (non-zero area) is infinity. Why? Because a point does not necessarily have to have integer coordinates - there can be a point at (0,0), (0,0.1), (0.001), (0,0.0001) and so on. I think what you mean by points in your question is that all points must have integer coordinates (i.e. lattice points), or alternately, "cells" in a rectangular grid (like cells on a chess board). Please let me know if I misunderstood your question.

There are 600 rows and 300 coloumns. This means that there are 600 * 300 = 180,000 different cells. It follows that there are nCr(180,000,2) = 16,199,910,000 unique pairs in the grid. I am assuming you consider the pair ((1,1),(2,2)) and ((2,2),(1,1)) equivalent. Otherwise, there are 180,000*180,000 = 32,400,000,000 pairs.