We have $k=3$ descriptors and for the sake of argument let's label them $-1,0,1$, each assigned to $3$ different pairs of attributes (thus in total $m=6$ attributes), say $(a_1,a_2)$,$(b_1,b_2)$,$(c_1,c_2)$.
This can be viewed as a single vector defining a different case of the model we are studying: $(a_1,a_2,b_1,b_2,c_1,c_2)$.
We wish to "weed out" the symmetric cases, for example the case $(0,1,-1,-1,0,0)$ is symmetric to $(1,0,-1,-1,0,0)$ because values that define the same attribute are interchanged. Similarly, the case $(1,1,-1,0,-1,1)$ is symmetric to $(1,1,0,-1,-1,1)$ and so on..
In total we have $3^6=729$ combinations, but this is significantly overcounting, because the symmetric cases have been counted twice.
How do we go about counting the number of distinct cases in general?
Best Answer
For each pair of attributes, a multiset of size $2$ will define the assignment of the descriptors for those attributes without overcounting.
Since there are ${{3+2-1}\choose{2}} = 6$ multisets of size 2 with elements taken from a set of 3 elements (descriptors), and there are three pairs of attributes to assign descriptors, the answer is
$$6^3 = 216$$
See https://en.wikipedia.org/wiki/Multiset#Counting_multisets for details on multisets.