The combinatorics of random walk with absorbing barriers

Consider a particle executing random walk on the line. The particlc starts at the point 1 and arrives eventual+ at the point N in a total of N + 211 1 equally probable unit steps, I of which are in the negative direction. Both the origin and the poini Iv are taken to be absorbing barriers, so that the particle may never visit 0, at: * may reach N only at the end of the walk. We seek the number PN,I of distinct walks satisfying these restrictions. In t;le third edition of his celebrated book. FelIer 14. p. 961 gives an explicit solution to this problem: