MHB What is the relationship between r-multisets and combinations of 0s and 1s?

[mathmaniac1]

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]

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.

See also Theorem 2 on this Wiki page.

 

[mathmaniac1]

Re: The nof r-multisets

Thanks for replying,E.M

but I had got it on my own.