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At sufficiently low packing fraction, neglect overlaps of the excluded regions around different disc centres. Sequential insertion gives the configurational integralTaking logarithms and expanding to first order in density,For large $N$, replace $N-1$ by $N$. The expression $N\ln (a-Nb)$ has the same leading-density expansion when $b=2a_d$. Thus the hard-disc excluded-area approximation yieldsThe factor $N!$ accounts for indistinguishable particles and ${\Lambda }^{2N}$ for two-dimensional momentum integration. The factor one-half between pair-excluded area and $b$ avoids counting every pair twice. Equality of leading expansions does not make this free-volume resummation exact at high density.