TITLE: Combinatorial proof for identity $\left(\!\!\binom{n\vphantom{1}}{k}\!\!\right)=\left(\!\!\binom{k+1}{n-1}\!\!\right)$ (multiset coefficients)
QUESTION [7 upvotes]: In class we have recently started using combinatorial proofs. I have tried this problem that our teacher has assigned as a "challenge". I understand how to receive the left hand side, but am struggling with the right. I do no see how the k is now on the top. I understand the use of the stars and bars technique but do not know how to apply it to this side. Any help would be highly appreciated. The problem reads:
Prove with a combinatorial proof:
$$\left(\!\!\binom{n}{k}\!\!\right)=\left(\!\!\binom{k+1}{n-1}\!\!\right)$$
where the notation means multiset choose.

REPLY [2 votes]: Let $N$ be the number of ways to put $n-1$ red balls and $k$ blue balls into a row. Each such configuration can be regarded 
(a)  as an encoding of a multiset of  cardinality $k$ on $\{1,2,\ldots,n\}$, using the $n-1$ red balls as separators,
and at the same time
(b) as an encoding of a multiset of  cardinality $n-1$ on $\{1,2,\ldots, k+1\}$, using the $k$ blue balls as separators.