On some problems concerning a restricted random walk

Extract

A particle that is located at the position (0, 0) in the (x, y) plane at the moment x = 0 is subjected to random impulses at the moments of time x = 1, 2, …, a + b. As the result of each impulse, the particle may be displaced either by a positive or by a negative step with respect to the y-axis; it is given that the particle is restricted by the condition that it has exactly a positive and b negative displacements. Hence in the (x, y) plane the path of the particle at each impulse is depicted by a shift of a unit to the right and by one unit upward or downward; that is the particle's path starts at (0, 0) and terminates at (a + b, a – b) after a + b steps. We assume that all possible different paths are equally probable, and assume also, without loss of generality, that a ≧ b.