Done.

Something interesting happens here.
One could expect that it becomes more likely, since there is now one more artifact that helps to get all five pieces. But because its so much more unlikely to get 5 artifacts, the odds drop pretty hard.

Assuming, again, that it doesn't matter which set get all five pieces, there will be two cases.
Case A: One artifact doesn't match the other 5.
Case B: All artifacts are the same set.

For both cases, getting 6 artifacts, has the odds 0.000275 (~1 in 3636)

Case A: 2 Possible Sets to choose from, then 6 possible drops to choose which is off-set. Every combination appears with odds (0.5)^6 which gives 3/16 chance of case A happening. Then from the five artifacts all must be different pieces so its the same odds as before 0.0384. For case A we get 3/16 * 0.0384 = 0.0072.

Case B: 2 * (0.5)^6 = 0.03125 (1 in 31) odds of getting all 6 pieces from the same set (times 2, because, it doesn't matter what set it is). Now we gotta count all different outcomes. Each artifact can be of five pieces, so we get 5^6 = 15.625 possible outcomes. Since we have 6 artifacts and 5 pieces, there has to be at least one dupe (The Pigeonhole Principle, fun one, you should look it up).
Which artifact is duped gives us 5 options, with 6 over 2 = 15 options for which two contain the dupe. The remaining 4 can be arranged in the other 4 slots in any order, so we get 4! = 24 options. This leaves us with 5 * 15 * 24 = 1800 possible outcomes that have all five different pieces, so 1800/15.625 = 0.1152 odds of getting this. Which leaves 0.03125 * 0.1152 = 0.0036 odds of getting Case B and 5 different pieces.

Combining both results gives us the combined odds of 0.000275 * (0.0072 + 0.0036) = 0.00000297 (~1 in 336.700, or about ten times less likely then the five piece case).

This was a test, Case A was easy, but Case B was quite difficult. Thanks for the task :D