A rigorous analysis of counting paths for 1-D random walk in the presence of a reflecting barrier is presented. A combinatorial proof is given showing that in presence of a barrier situated at m=o the number of paths departing from m=2j and arriving at m=2k on the positive x-axis after 2M steps is given by a(2j,2k,2M)=C(2M, M-j+k)-C(2M-j-k-1) where C(n,m) denotes the binomial coefficient. Likewise the total number A(2j,2M) of all paths departing from m=2j is given by A(2j,2M)=C(2M,M)+2SUM(i=0,j)C(2M,M+i). These formulae enable to calculate any random walk redistribution of a diffusing species near or at the reflecting barrier. Based on this analysis it is shown that for a particle starting its random walk near or at the barrier, the probability of finding it at the interface is diminishing with the number of diffusional steps as 1/(M+1) and that the peak of the probability distribution is moving away from the barrier with the increasing number of steps N as SQRT(2N). Thus the subsurface region is progressively depleted. The present analysis has bearing on the treatment of diffusion of impurities and point defects, and of entropy in thin films and in sub-surface layers.