 Solution

With agent variables, each constraint domain has cardinality $2^m$. Their total number of listed domain values is $n{2}^m$, while the Cartesian product of the domains hascomplete assignments before constraints.

With item variables, each constraint domain has cardinality $n$. Their total number of listed domain values is $mn$, while their Cartesian product hascomplete assignments. These are raw domain-space sizes. The larger agent-variable space includes overlaps and missing items; imposing the allocation constraints reduces it to the same $n^m$ actual complete allocations represented directly by item variables.