= Solution
Freeze neighbouring particles and approximate the mobile particle's accessible area by $a_c$. Its one-particle translational contribution to the <canonical partition function> is $a_c/\Lambda^2$ in two dimensions. Neglecting correlations between different cages,
$$
\boxed{Z_{\rm cell}\simeq(a_c/\Lambda^2)^N,\qquad F=-Nk_BT\ln(a_c/\Lambda^2).}
$$
There is no additional uncancelled $N!$: assigning particles to cages removes permutations, while restoring all $N!$ assignments cancels the indistinguishability factor. The result is a free-volume approximation; collective cage motion is excluded by construction.
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