  We present exact calculations of the average number of connected clusters per
site, $<k>$, as a function of bond occupation probability $p$, for the bond
percolation problem on infinite-length strips of finite width $L_y$, of the
square, triangular, honeycomb, and kagom\'e lattices $\Lambda$ with various
boundary conditions. These are used to study the approach of $<k>$, for a given
$p$ and $\Lambda$, to its value on the two-dimensional lattice as the strip
width increases. We investigate the singularities of $<k>$ in the complex $p$
plane and their influence on the radii of convergence of the Taylor series
expansions of $<k>$ about $p=0$ and $p=1$.
