= Solution
The <three-point density correlation of an ideal quantum gas> follows first for a <number-conserving Gaussian quantum state>. There are six pairings, one for each <permutation> of three annihilation operators. The identity gives the product of densities; each <transposition> has sign $\epsilon$; both three-cycles have positive sign. Thus
$$
\boxed{\begin{aligned}
C^{(3)}_{123}={}&n_1n_2n_3
+\epsilon\left[n_1|g_{23}|^2+n_2|g_{13}|^2+n_3|g_{12}|^2\right]\\
&+g_{12}g_{23}g_{31}+g_{13}g_{32}g_{21}.
\end{aligned}}
$$
The last two terms are <complex conjugates>. They equal $2\operatorname{Re}(g_{12}g_{23}g_{31})$, not an additional signed pair term. This is the <permanent of a matrix> $(g_{ij})$ for <bosons> and its <determinant> for <fermions>. <Wick's theorem> supplies the six terms only with the state assumption stated in part (a).
For completeness an exact bosonic Fock-state alternative can be given without that assumption. Let $\boldsymbol\ell$ be a mode-occupation decrement with $\sum_\alpha\ell_\alpha=3$. Define $A_{\boldsymbol\ell}(1,2,3)$ as the sum of $\prod_{j=1}^3\varphi_{\alpha_j}(\mathbf r_j)$ over all distinct ordered triples having these mode multiplicities. Acting three annihilation fields on the <number state> leaves mutually orthogonal states for different $\boldsymbol\ell$. Therefore
$$
\boxed{C^{(3),\rm Fock}_{123}=\sum_{\boldsymbol\ell:\,\sum\ell_\alpha=3}
\left[\prod_\alpha N_\alpha^{\underline{\ell_\alpha}}\right]
|A_{\boldsymbol\ell}(1,2,3)|^2}.
$$
Here $N^{\underline\ell}=N!/(N-\ell)!$, with zero when $\ell>N$. This expression includes repeated-mode occupation corrections absent from the Gaussian permanent; for a single mode it reduces to $N(N-1)(N-2)\prod_j|\varphi(\mathbf r_j)|^2$.
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