The number of different multisets There are ${n \choose k}$ ways of choosing $k$ distinct items from a set of size $n$. There  are $n^k$ ways of choosing $k$ items if we allow duplicates. How many distinct multi-sets are there of size $k$?  
It must be considerably fewer than $n^k$ as, if the set $S = \{1,2,3,4,5\}$ then we say the multi-subsets $\{1,1,2,3\}$ and $\{1,2,1,3\}$, both of size $4$, are the same.

It is called multiset coefficient. 
$$\left(\!\!{n\choose k}\!\!\right)=\binom{n+k-1}{k}$$
See here http://en.wikipedia.org/wiki/Multiset_coefficient#Counting_multisets
