As far as I know not even in Mathematica there is a predefined function that solves the problem. Strictly speaking, the problem can't be solved: randomly placing circles finally asks for overlap. If you have a strategy to avoid this then your distribution of disks is no longer random. There are, however, workarrounds that may turn out to be saisfactory in practical situations:

1: Generate random circles sequentially. If a circle fits (does not overlap with any circle in place) we place it and try a new one else. Soon we reache a point where we need prohibitively many attemts for finding a free place. We then stop and accept that we probably got less circles than it was our initial intent. 

2. We place the random circles irrespective of overlap and implement an idealized particle mechanics which lets overlapping (or nearly overlapping) disks feel a repelling force so that in a time-stepped  mechanical algorithm the 'particles' arrange in an non-overlapping configuration. This is the strategy with which I gathered some experience and where I could help with proven algorithms.

3. We divide the available space into rectangular cells where a cell is reasonably larger than the chosen circles. Then place into each cell a circle and select the position randomy in the available cell-space. This is the simplest effective method but it fails to model the kind of clustering which is typical for randomly arranged items.

A 1000 * 1000 dimension can only contain certain number of distinct non-overlapping circles. If the circles positions are not random, you could just vary the radius(pixels) such that you may have more than 20 non overlapping circles.

For random circle positions, still you have to use an appropriate fixed radius. Just monitor each circle's :

This should prevent overlapping, I hope it helps?

A couple of years ago I've written a simulation

of falling balls or falling circles, which could

be stones or granular material, or atoms,

a sedimentation process. The idea was to vary

the positions of the circles in steps randomly

to minimize potential energy as well as

overlapping energy. I omitted kinetic energy.

Overlapping energy was defined as a constant

times square of overlapping length. If two

circles don't overlap, the overlapping

energy is defined as 0.

http://www.augos.com/temp/Kugeln.zip

I know there are more efficient algorithms

to determine whether two circles are very

close, like quadtrees. But for the sake

of simplicity I handled all pairs of

circles.

Regards,

Joachim

It seems to be ill ocnditioned question. Either radius or ratio dimension/ radius should be amended.

Something more. Time constraints, lower or upper number of circles to be placed.

First, any particular programming language has very little to do with stated problem.  When you know what you really want, you'll find the way to program it in your favorite (and well mastered) computer language.

Now to the point.  If your desired result is random placement of non-intersecting circles with fixed radii (I assume they all are equal) then your practical observation shows what is possible (a phenomenon called percolation comes to the mind).  But, perhaps, you mean "uniform" rather than "random"?  If this is the case, then Ulrich's or Joachim's advices seem reasonable at least.  Yet, there is another possibility, if you insist on the word "random".  Simply generate only the positions of centers of your future circles, as many as you need.  After this is done, all you have to do is to inflate those points to full-blown identical circles.  The radii of these circles should be set, of course, as not exceeding  half of the shortest distance between initial points.