161827

Ordered list: a, b, 3, c, d. {a, b, c, d} all positive integers.

Fact 1: a≤b≤3≤c≤d, to keep list ordered.

Fact 2: a+b+c+d=17, to make the average 4, keeping 3 in the middle.

Case 1:
If a=b=c=3
d must be 8, see Fact 2.

Case 2:
If b=c=3
a+d=11 (so fix average at 4), a<3<d.
Follow rule until you exhaust the possibilities.

Case 3:
If b=3
a+c+d=14
a<3<c<d, c≠d (as this would make the set bimodal)
Follow this rule until all possibilities are exhausted.

Case 4:
If c=3
a+b+d=14
a<b<3<d, a≠b (as with Case 3)