The electron counting (EC) rule helps explain semiconductor surface stabilization and favored adsorbate coverages, but does not select the preferred arrangement when many configurations satisfy it. Motivated by the idea that satisfying the rule locally favors stability, we enumerate configurations of GaN(0001)-$(6\times 6)$ with 3 Ga adatoms and 18 H atoms that admit decomposition into EC-balanced $(2\times 2)$ blocks. Even this restricted space contains 416,683 symmetry-distinct configurations, making energy-based searches costly. We introduce the tiling decomposition multiplicity, $n_{til}$, which counts the compatible decompositions, and evaluate it for every configuration. Among 2,529 configurations relaxed with a DFT-validated machine-learning interatomic potential, the lowest energy in 13 of the 14 Ga placement symmetry classes is attained by a configuration with maximal $n_{til}$ within that class. In the remaining class, a high-multiplicity configuration lies 8.5 meV below the maximizer (17.5 meV in DFT), well under $k_{B}T$ at growth temperature. The energy ordering is largely associated with avoiding adjacent bare surface Ga sites: each pair costs about 0.2 eV. The existence of a compatible decomposition is a separate global requirement. The 24 multiplicity maximizers are exhaustively identified and all evaluated with DFT. Counting compatible decompositions thus narrows 416,683 configurations to 24 promising candidates for first-principles evaluation, without computing the energy of every configuration.