 Solution

At coincident points the Gaussian expression has $g_{ij}=n$. Its six contributions giveWhen all mutual separations become large and the density stays the same nonzero $n$, only $n_1{n}_2{n}_3=n^3$ remains. ConsequentlyThe bosonic value is $3!$ from Gaussian exchange pairings; the fermionic determinant vanishes when its rows coincide. Density-normalized correlations avoid the additional homogeneity assumption. For a specified bosonic number product, part (c)'s occupation sum must replace the six-pairing formula, so neither the universal factor six nor a universal printed ratio follows from that specification alone.