Barium carbonate is more soluble in water than in aqueous sodium carbonate because?

The answer to your question is Choice(c). Remember that BaCO3 is a weak electrolyte and a sparingly soluble salt whereas Na2CO3 is a strong electrolyte and highly soluble in water. Now, in an Na2CO3 solution as CO32- ion is common between the two so ionization of the weak electrolyte BaCO3 will get suppressed. In only water there isn't any common ion effect so solubility of BaCO3 will be better. Choice(a) is incorrect as there isn't any question of displacement here. In Choice(b) this is totally incorrect as Lots of barium salts like BaCl2, BaF2, Ba(OH)2 etc are highly soluble. And lastly in choice(d) BaCO3 is an ionic molecule and not covalent. The correct choice is (c).