Number of possible 2- and 3-letter combinations from a set of 6, disallowing certain letters to be used together?

Question

Say we have a set of 6.

 

A

B

C
X

Y

Z

 

These letters are arranged in a hexagon. We can have the letters in subsets of 1, 2, or 3, but the same letter cannot repeat in a subset, and each letter cannot be in a subset with its "opposite" - A cannot appear with X, B cannot appear with Y, and C cannot appear with Z.

 

For example, we can have ABC, or BX, or XY, but not BCZ, or ABX, or BY.

 

My question is, how do I figure out how many possible answers there are with these rules? There are 6 by default, as singular letters (A, B, C, X, Y, and Z).

 

The only method I've come up with is calculating the number of possible combinations (I believe there are 120 3-letter combinations, without repetition) and manually removing any entries that do not match the opposite rule.

Answer 1

It seems to me that for sets of size 3, you can only have 3 letters that are right next to each other in the hexagon so there would only be 6 possible combinations: ABC, BCX, CXY, XYZ, YZA, ZAB

 

For sets of size 2 it seems easy enough to count them manually, once you are done with a letter you don't need to worry about using it again:

 

AB, AC, AY, AZ

BC, BX, BZ

CX, CY

XY, XZ

YZ

 

So 12 total combinations for those.