  We report a method that uses "completeness" to estimate the number of
exrasolar planets discovered by an observing program with a direct-imaging
instrument. We develop a completeness function for Earth-like planets on
"habitable" orbits for an instrument with a central field obscuration, uniform
sensitivity in an annular detection zone, and limiting sensitivity that is
expressed as a "delta magnitude" with respect to the star, determined by
systematic effects (given adequate exposure time). We demonstrate our method of
estimation by applying it to our understanding of the coronagraphic version of
the Terrestrial Planet Finder (TPF-C) mission as of October 2004. We establish
an initial relationship between the size, quality, and stability of the
instrument's optics and its ability to meet mission science requirements. We
provide options for increasing the fidelity and versatility of the models on
which our method is based, and we discuss how the method could be extended to
model the TPF-C mission as a whole, to verify that its design can meet the
science requirements.
