This is not that hard to do manually. I bet it takes you longer to research a program or build one than to just do it by hand.
Make a PowerPoint (or other drawing program that is the urn. Big enough to hold two stacks of at least 9 balls. You will have the blue ones always on the left and red ones always on the right. In separate stacks (not mingled as you showed here).
I assume the urns don't themselves need to follow some distribution (like stat mech microcanonical ensembles). So write out the urns in terms of number of blue and number of red:
All the combinations of that give 100 choices (you do have one empty set, but if that is a problem just add the two 10-0 combinations and have 101 choices.
Make PPT drawing objects of 0-9 blue balls (stack them into 3-3). Use the group command (not critical for blue) and save a slide with each choice on it (in the urn). Now duplicate each slide 10 times (in slide sorter view with cut and paste command). You will then have a 100 slide presentation. When you do the copy do it 10 slides at a time (iow sort of "collated"...not 0 balls 10 times and then 1 ball 10 times, but ten times of 0, 1, 2, etc.
Create the ten different red ball objects (just make ten more slides at end of your presentation or make a separate presentation for the red ball objects. Here you do need to use the group command for the drawing object, because you will be cutting and pasting the objects into the 100 urns. Go one at a time and paste each of the ten objects into ten slides.
Probably take you a couple hours max.