= Solution
For $k_{\rm B}T\gg\delta$, the <Boltzmann factors> of the orientations become almost equal, giving $3^N$ nearly equiprobable <microstates>. The <Boltzmann entropy> of that distribution approaches
$$
\boxed{S_{\max}=Nk_{\rm B}\ln3.}
$$
\b[The exact limiting value is reached only as $T\to\infty$] for a nonzero splitting. The useful crossover scale is $T\sim\delta/k_{\rm B}$; one needs $T\gg\delta/k_{\rm B}$ for a good saturation approximation. The binding offset $\varepsilon$ is not an additional population scale here because all sites remain occupied and it shifts every orientation equally.
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