With only 12 combos I would test them all, but when the number gets much higher it's time to just sample them. Unless the selections are super critical then select maybe the first, last and one or two in the middle. Just remember, the more you don't test the higher the risk of a defect in the field.
Testing is never finished, just abandoned!
Success is the ability to go from one failure to another with no loss of enthusiasm.
~ Winston Churchill ~
If you have a complex environment, you have to optimize the test.
Imagine you have a million combinations and it takes a second to run only one of them. Then the whole test will run for 11 days. Is it acceptable and necessary in your environment? If yes, no actions need to be done.
If no, you can try to either reduce time spent for each combination, or to reduce total number of cases, or both.
To reduce the time spent for each test case, you can separate testing of back-end and front-end. For example, you have 6 options by 10 variants each. Here you can divide tests on two groups. One of them will test that part of back-end for each this 10^6 combinations are critical and one will test front-end where it may be acceptable to test only that front-end executes proper option from back-end.
If you go further, and find parts of tested system that can be tested separately, you can reduce the number of test cases themselves.
Another method for reducing combinations is called “All-Pairs”. James Bach wrote a tool to generate such combinations. You can take it from http://www.satisfice.com/tools.shtml
It includes the description of the method. Using this method you can reduce number of cases down to several hundreds or few thousands.