I tried to combine all three conditions at one place.
In cell E2 CTRL+SHIFT+ENTER this formula and not just ENTER and then copy down:
=LOOKUP(SUM(--(A2:D2*(1/TRANSPOSE(A2:D2))=1)),{4,6,8,10,16},{"Case 1","","Case 2","Case 3",""})
Case 1 >> All 4 different
Case 2 >> 2 Pairs
Case 3 >> One Number appears 3 times
2 pink marked empty slots are:
first one is 'at least one pair' and second one is all 4 match just in case if you decide to chase them.
Once this is done then it is plain COUNTIF formula for your cases. e.g.
=COUNTIF($E$2:$E$32,"Case 1")
I'd think someone from statistical background can give you a better formula than the above one.
e.g.
Normally entered:
=MAX(COUNTIF(A2:D2,MODE(A2:D2)),1)
will give you maximum matching numbers starting from 1 (no match) to 4 (all four match).