The number of r-multisets with n distinct objects is equal to the number of ways to arrange r zeros and (n-1) ones. This relationship is established by interpreting r-multisets as tuples (r1, r2, ..., rn) where each ri represents the count of distinct objects, satisfying the condition that the sum of all ri equals r. Arranging r zeros in a row with n-1 ones inserted between them effectively divides the zeros into n groups, which corresponds directly to the formation of r-multisets.
PREREQUISITESMathematicians, computer scientists, and students studying combinatorial mathematics or set theory, particularly those interested in the applications of multisets and combinatorial arrangements.
f r-multisetsI assume that $r$-multisets means multisets of cardinality $r$, and by combining zeros and ones the problem means arranging them in a row. The number of multisets of cardinality $r$ consisting of $n$ distinct objects equals the number of tuples $(r_1,\dots,r_n)$ such that $r_i\ge0$ for all $i$ and $\sum_{i=1}^n r_i=r$. Here $r_i$ serves as the number of copies of object number $i$. In turn, arranging $r$ zeros in a row and inserting $n-1$ ones between them amounts to dividing zeros into $n$ groups, some of which may be empty. Thus, each arrangement gives rise to a tuple described above.mathmaniac said:
f r-multisets