Random User Pairing

You are given an array of user identifiers. Write a function that produces a set of random pairings from this list, subject to the condition that every possible complete pairing configuration is equally likely. The process must obey the following rules:

• Two distinct users are selected at random to form a pair, and this continues until fewer than two users remain.
• The selection procedure must be uniformly random: each valid collection of pairs that could be generated from the input must have the same probability of occurring.
• When the input contains an odd number of users, exactly one user will be left without a partner.

Example 1:

Example 2: