It's a pretty straight forward set of displacement reactions occurring. I got this question wrong too though. Basically each time you add a chemical to the precipitate a displacement occurs because what you're adding is LESS soluble than what you had. So SO4 is initially displaced by I-, and so the precipitate is now PbI2, meaning I- is less soluble than SO4 (because it precipitated out while SO4 dissolved in solution). This happens again with CO3, it displaces I- and precipitates out. This means that CO3 is less soluble than I-, which is less soluble than SO4. So they would precipitate out, with the least soluble thing coming first, CO3, I-, then SO4.