If you have the budget you want Trapcode Particular for this project. If you must use what comes with AE I'd use Card Dance or duplicate a couple of animated comps where you just animated the dollars falling down and then drop these slightly different animations in a main comp and duplicate them and move them around and adjust the timing of them until you have the look you want. If you have no idea how to use Particular creating five or six comps with falling bills and then duplicating and moving the duplicates in x would probably be the fastest.
There are several animation presets out there that will automatically cause a layer to fall, even bounce, and then come to rest. You might also consider looking at some of them. If you want to go that way let us know and we'll try and provide links to the animation presets or scripts.
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Inter-particle collision detection/ accumulation is not part of any AE plug-in I know of, not the ones that come with AE nor any commercial ones. The math is too complex for reasonably realtime performance. that said, even in a 3D program thin layer collision objects are extremely complicated to get right, so simply do what Rick said - fake it with whatever means you have at hand, including mass duplicating and retiming a pre-comped animation.