We use \(N_{t}\), the number of exoplanets observed in time \(t\), as a science metric to study direct-search missions like Terrestrial Planet Finder. In our model, \(N\) has 27 parameters, divided into three categories: 2 astronomical, 7 instrumental, and 18 science-operational. For various "27-vectors" of those parameters chosen to explore parameter space, we compute design reference missions to estimate \(N_{t}\). Our treatment includes the recovery of completeness \(c\) after a search observation, for revisits, solar and antisolar avoidance, observational overhead, and follow-on spectroscopy. Our baseline 27-vector has aperture \(D = 16\)m, inner working angle \(IWA = 0.039''\), mission time \(t = 0-5\) years, occurrence probability for earthlike exoplanets \(\eta = 0.2\), and typical values for the remaining 23 parameters. For the baseline case, a typical five-year design reference mission has an input catalog of \(\sim\)4700 stars with nonzero completeness, \(\sim\)1300 unique stars observed in \(\sim\)2600 observations, of which \(\sim\)1300 are revisits, and it produces \(N_{1}\sim50\) exoplanets after one year and \(N_{5}\sim130\) after five years. We explore offsets from the baseline for ten parameters. We find that \(N\) depends strongly on \(IWA\) and only weakly on \(D\). It also depends only weakly on zodiacal light for \(Z 0.2\), and scattered starlight for \(\zeta < 10^{-10}\). We find that observational overheads, completeness recovery and revisits, solar and antisolar avoidance, and follow-on spectroscopy are all important factors in estimating \(N\).