TLDR; The formula is n(n-1)/2 where n is the number of items in the set.
Explanation:
To find the number of unique pairs in a set, where the pairs are subject to the commutative property (AB = BA), you can calculate the summation of 1 + 2 + ... + (n-1) where n is the number of items in the set.
The reasoning is as follows, say you have 4 items:
A
B
C
D
The number of items that can be paired with A is 3, or n-1:
AB
AC
AD
It follows that the number of items that can be paired with B is n-2 (because B has already been paired with A):
BC
BD
and so on...
(n-1) + (n-2) + ... + (n-(n-1))
which is the same as
1 + 2 + ... + (n-1)
or
n(n-1)/2
