The multisubset sum problem for finite abelian groups

We use a similar techique as in M. Kosters, The subset problem for finite abelian groups, J. Combin. Theory Ser. A 120 (2013), 527-530, to derive a formula for the number of multisubsets of a finite abelian group G with any given size and any given multiplicity such that the sum is equal to a given element g from G. This also gives the number of partitions of g into a given number of parts over a finite abelian group.