# The Number of r-multisets

#### mathmaniac

##### Well-known member
The no.of r-multisets with n distinct objects is
equal to the number of ways of combining r-0s and (n-1) 1s.

How?Can anyone explain this?

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
Re: The nof r-multisets

The no.of r-multisets with n distinct objects is
equal to the number of ways of combining r-0s and (n-1) 1s.
I 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.