An Algorithmic and Recursive Approach to Combinatorial Occupancy Problems with Repetition

This paper presents an autonomous, constructive algorithmic framework to evaluate the exact probability distribution governing the retrieval of multiple unique outcomes within a finite sequence of independent trials with replacement. By tracking the sequential microscopic branching of extraction strings through an evolutionary tree, we define structural isomorphic string families and derive the fundamental finite difference recurrence relation that dictates the occupancy framework under a strict logical AND condition. The proposed model bypasses the massive, computationally intensive loops of traditional static combinatorics by employing positive integer recursive matrices. To bridge local target requirements with global sampling domains, we introduce asymmetric group projection multipliers that filter macro-state expansions through their exact geometric weights, resolving overlapping subset boundaries with absolute precision. The mathematical stability and algebraic integrity of the algorithm are verified through a complete numerical application, demonstrating perfect convergence with the classical Principle of Inclusion-Exclusion. This recursive architecture establishes a highly efficient, computationally optimal layout for parallel programming loops, enabling exact statistical mapping of large-scale sampling distributions.