A discrete time correlated random walk in one dimension is investigated. Combinatorial arguments are used to calculate the exact probability distribution P-N(L), the probability that the correlated random walker arrives at a distance L steps to the right of its starting point after exactly N steps. P-N(L) is calculated using arbitrary initial conditions which permit the influence of end effects and boundary conditions to be calculated and several special cases are considered in detail. P-N(L) with arbitrary initial conditions is calculated both with and without a bias for motion in one direction yielding a useful model for the combined diffusion and drift of charged particles undergoing a correlated random walk in an applied field. The relation of the correlated random walk to the Ising model is also discussed.