From ChatGPT – which solution is better? :)
"Pairwise disjoint" means that no two sets have a common element, i.e., A ∩ B = ∅, B ∩ C = ∅, and A ∩ C = ∅.
The problem is a bit tricky since the subsets are chosen randomly. In general, the probability can be quite complex to calculate and may involve advanced combinatorial techniques.
But if we simplify things and suppose that each element of X independently decides which of the sets A, B, and C to go into, then the answer is relatively simple:
1) Each element has four choices: to go into none of the sets, or to go into one of A, B, or C. So there are 4^n ways to distribute n elements into {A, B, C, ∅}.
2) If A, B, and C are pairwise disjoint, then each element has two choices: either go into ∅ or go into one of the sets {A, B, C}. So there are 2^n ways to distribute n elements into disjoint {A, B, C, ∅}.
Therefore, the probability that A, B, and C are pairwise disjoint is (2^n) / (4^n) = (1/2)^n.
Please note: This is a simplification that may not apply if the subsets A, B, and C are not chosen independently or if other constraints are in place. Sometimes, the only feasible way to find the solution is to use more sophisticated combinatorial methods.