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 Ly, of the square, triangular, honeycomb, and kagomé lattices Λ with various boundary conditions. These are used to study the approach of k , for a given p and Λ, 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.