How to build up the answer as we go:
Cade has six red stamps and three blue ones.
Thus Cade must have at least 9 stamps.
In his collection, seven stamps are from Argentina and six stamps are from Sweden.
This means he has at least 13 stamps, with 7 from Argentina and 6 from Sweden. Any three of them are blue and any other six of them are red.
One stamp is purple and it is not from Argentina or Sweden.
Now he must have at least 14 stamps: 7 from Argentina and 6 from Sweden, and one from Elsewhere.
Two of his Argentine stamps are red and one is blue.
This is where it gets tricky. Among the six red stamps, only two can be from Argentina. The other four can be from Sweden. Among his three blue stamps, one can be from Argentina. Let's assign the other two to Sweden. Now we still have 14 stamps:
2 Argentina/red, 1 Argentina/blue, 4 Argentina/x, 4 Sweden/red, 2 Sweden/blue, 1 Elsewhere/Purple.
Two of his Swedish stamps are blue and three are red.
Luckily, we had assumed that there were two blue Swedish stamps. However, we need to take one red one away from Sweden.
Our configuration after this rule: 2 Argentina/red, 1 Argentina/blue, 4 Argentina/(not red or blue), 3 Sweden/red, 2 Sweden/blue, 1 Elsewhere/Purple, and 1 Elsewhere/Red.
Since every group has either a different color or a different source location, we can't combine any groups. There may be more stamps, but all of the stamps in the rules have been accounted for exactly once. Thus, the rules indicate a minimum of 15 stamps.