Bounded compositions and products of chains, or bi(s)nomials everywhere from de Moivre's formula to spin multiplicities and constant-sum codes

How many ways are there to select $k$ objects from $n$ types when type $i$ is available in only $s_{i}$ copies? For equal bounds, this is de Moivre's problem of obtaining a prescribed total with $n$ dice, now a textbook application of the principle of inclusion-exclusion. For arbitrary bounds, the a