Direct Closed-Form Cardinalities for Bounded Multisubset Classes

This paper provides a direct, non-recursive mathematical framework to determine the exact cardinality of multisubsets derived from a regular parent multiset M=(r^n). Extending the combinatorial foundations of Singh and Singh (2003), we partition the submultiset space into eight distinct, disjoint classes using element repetition (R/NR) and identity/order (B/NB) configurations. We derive explicit, closed-form formulas for each and all of these eight classes. Using only the parent parameters n and r, these formulas instantly calculate class cardinalities, bypassing computationally heavy generating function expansions and manual partition groupings.