  Thermodynamics studies of a prototypical quasi-two-dimensional frustrated
magnet Ba$_2$Sn$_2$ZnCr$_{7p}$Ga$_{10-7p}$O$_{22}$ where the magnetic Cr$^{3+}$
ions are arranged in a triangular network of bipyramids show that the magnetic
zero-point entropy for $p=0.98$ is 55(1)\% of the entropy expected when the
Cr$^{3+}$ moments are fully disordered. Furthermore, when combined with a
previous neutron scattering study and the perimeter scaling entropy of a spin
jam, the analysis reveals that with decreasing $p$, i.e., doping of the
nonmagnetic Ga$^{3+}$ ions, the variation in the magnetic zero-point entropy
can be well explained by the combined effects of the zero-point entropy of the
spin jam state and that of weakly coupled orphan spins, shedding light on the
coexistence of the two types of spin states in quantum magnetism.
