Firstly, create the initial design via the ellipse tool. You can apply it once and then use the alt / option and duplicate or just create a new ellipse (hold the shift down to create a circle) and repeat that.
You can fill the art board with the same color etc but say you want to only use red, green and blue. Select one of the paths and apply the red and go to swatches panel and right side menu and new swatch (you may like to delete all the other swatches, note that this is only for the current document and as soon as you open a new document, they will all be back as before). Go to another path and set that to green and repeat the swatch command and then go to another path and set that to blue and repeat the swatch command
You should now have three swatches and this is useful to keep everything consistent plus more useful when you come to some of the other commands mentioned later.
Select many or all the paths and then hold down the alt / option key and drag and duplicate those and then scale that selection and perhaps rotate and shift to different position. Select a different set of paths and repeat the duplicate and scale and rotate and so on (if you want them all the same size then don't re-size them in any way). You can then rapidly fill the entire art board with your design. Perhaps select a hundred paths and repeat this and within a few seconds, you will have a thousand paths with red / green / blue (or any other combinations)
Go to the edit menu and edit and then use the preset option and library and with that you have to choose a swatches set to work with and I would suggest the document swatches as you have three already in there (of course, if you have more than that then they will be used as well). Click OK and then go to the middle of the panel and use the randomly change the order so if you have a set of red designs selected and you click this button, they could become red again but equally they could go green or blue (or whatever you are using in the swatches panel). Of course, you can tweak these so they can be a totally different set such as yellow, orange and purple and you can do that by going to the sliders at the bottom of the panel and tweak the red into yellow etc and those settings will be used by the reorder.
Another option would be to select all the designs and then go to the Astute Graphics' plugin called Randomino (check out their website for freebies, demos, tutorials etc) and this randomizes loads of things but select the default top item and select the from list option. Go to the swatches panel and select the three presets there and then click the swatches button in the Randomino plugin and those will be added to the list, remove any others that are in the list. You can then click the randomize button at the bottom of the panel and it will just use those (red, green, blue etc) and will change all the paths
Another option would be to drag the paths (red, green etc) into the symbols panel, one symbol for the red, one for the green etc (or whatever you are using). Go to the sprayer tool and press return and set the intensity to 1 and the set density to 1, otherwise it will generate too many designs at once. Select one of them. Apply across the entire art board. Select another symbol and apply and then repeat with the last one (red, green etc). Double click the preset in the panel and you can then change it from red to blue or yellow etc and all the instances will change on exiting editing mode.