  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 $Z < 50$ zodis, end-to-end efficiency for $h > 0.2$, 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 $N$.
