  Continuous time random walks (CTRW) on finite arbitrarily inhomogeneous
chains are studied. By introducing a technique of counting all possible
trajectories, we derive closed-form solutions in Laplace space for the Green's
function and for the first passage time probability density function (PDF) for
nearest neighbor CTRWs in terms of the input waiting time PDFs. These solutions
are also the Laplace space solutions of the generalized master equation (GME).
Moreover, based on our counting technique, we introduce the adaptor function
for expressing higher order propagators (joint PDFs of time-position variables)
for CTRWs in terms of Green's functions. Using the derived formulae, an escape
problem from a biased chain is considered.
