hcbiochem
The way to approach this kind of problem is to consider all possible compounds and see if you can figure out which is not soluble. Now, this assumes that you have spent some time with the solubility rules that you have in your next and/or notes.
Aside from the two reactants, both of which are soluble, the two possible products formed by a double displacement reaction would be:
NH4NO3 and
Ag2CO3
One of the solubility rules is that all nitrate salts are soluble, so NH4NO3 will be soluble.
Another of the rules might be that most carbonates are insoluble or that most silver salts are insoluble. So, Ag2CO3 is an insoluble compound and will precipitate here.