A standard multiset coefficient, counting the number of multisets of size $k$ drawn from $n$ elements, is known to be $ \left( \! \binom{n}{k} \! \right) = \binom{n+k-1}{k}. $
Suppose there is a cap on the multiplicity of each element. For instance, suppose an urn contains 11 red balls, 6 green balls, 2 black balls, and 4 blue balls. How many ways can you draw a "hand" of 5 balls?
In general, suppose we use $\left( \! \binom{n}{k} \! \right)_{r_1, \dots, r_n}$ to denote the number of size $k$ multisets taken from $n$ elements in which, for each $j=1, \dots, n$, the multiplicity of element number $j$ is at most $r_j$. Is there a standard name and notation for this? Is it reducible in closed form to familiar combinatorial entities like binomial coefficients, factorials, etc?
