= Solution
In the <Asakura-Oosawa model>, nonadsorbing <polymers> are treated as ideal, mutually penetrable particles, but their centers cannot enter a layer around a hard <colloid>. The <polymer> <osmotic pressure> is $\Pi=\rho k_{\rm B}T$ in the ideal dilute limit, with the appropriate number density in the space dimension considered.
For fixed <polymer> number, reducing the inaccessible volume increases the available volume and lowers the <polymer> <Helmholtz free energy>. Equivalently, at fixed <polymer> <chemical potential>, the <polymer> contribution to the effective colloid potential is
$$
\boxed{U_{\rm dep}=\Pi\,\Delta V_{\rm excluded}=-\Pi V_{\rm overlap}.}
$$
When the excluded layers overlap, their union occupies less space than two separate layers. More <polymer> configurations become available, so the resulting <depletion force> attracts the colloids. It is an <entropy>-driven effective interaction, even though each microscopic colloid–<polymer> interaction is purely repulsive.
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