It is widely appreciated that balanced excitation and inhibition are necessary for proper function in neural networks. However, in principle, balance could be achieved by many possible configurations of excitatory and inhibitory synaptic strengths and relative numbers of excitatory and inhibitory neurons. For instance, a given level of excitation could be balanced by either numerous inhibitory neurons with weak synapses or a few inhibitory neurons with strong synapses. Among the continuum of different but balanced configurations, why should any particular configuration be favored? Here, we address this question in the context of the entropy of network dynamics by studying an analytically tractable network of binary neurons. We find that entropy is highest at the boundary between excitation-dominant and inhibition-dominant regimes. Entropy also varies along this boundary with a trade-off between high and robust entropy: weak synapse strengths yield high network entropy which is fragile to parameter variations, while strong synapse strengths yield a lower, but more robust, network entropy. In the case where inhibitory and excitatory synapses are constrained to have similar strength, we find that a small, but non-zero fraction of inhibitory neurons, like that seen in mammalian cortex, results in robust and relatively high entropy.