(3) Recall that a lattice path on the grid from (0,0) to (n, n) can be viewed as a sequence of length 2n which contains n-many R's and n-many U's. Let C. be the number of such paths which never go above the diagonal, i.e., those sequences in which there are never more U's than R's in any initial segment. Show that C, is equal to the number of ways of placing n unlabelled balls inn labelled urns so that for each k with 1sksn there are at most k-many balls total in the first k-many urns.

(3) Recall that a lattice path on the grid from (0,0) to (n, n) can be viewed as a sequence of length 2n which contains n-many R's and n-many U's. Let C. be the number of such paths which never go above the diagonal, i.e., those sequences in which there are never more U's than R's in any initial segment. Show that C, is equal to the number of ways of placing n unlabelled balls inn labelled urns so that for each k with 1sksn there are at most k-many balls total in the first k-many urns.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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