Counting isomers by trans pairs

You will count the geometrical isomers of an octahedral complex with the trans-pair method. The six corners form three pairs of opposite corners. An isomer is fixed by which ligands sit opposite each other.

The idea. Corners 0 and 1 are opposite, corners 2 and 3 are opposite, and corners 4 and 5 are opposite. Neither the order inside a pair nor the order of the pairs changes the isomer.

Your tasks.

1. Complete trans_key(order) so that it returns the three trans pairs as a sorted tuple of strings, each pair written in sorted order. For ("A", "B", "A", "C", "B", "C") the answer is ("AB", "AC", "BC").
2. Complete count_isomers(ligands) so that it returns the number of different keys over every order of the ligands.
3. Complete identical_pairs(key) so that it returns how many trans pairs hold two identical ligands, such as "AA".