= Solution
At coincident points the Gaussian expression has $g_{ij}=n$. Its six contributions give
$$
C^{(3)}(\mathbf r,\mathbf r,\mathbf r)
=(1+3\epsilon+2)n^3.
$$
When all mutual separations become large and the density stays the same nonzero $n$, only $n_1n_2n_3=n^3$ remains. Consequently
$$
\boxed{\text{bosons: }6,\qquad\text{fermions: }0}.
$$
The 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.
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