Consider the set of lattice paths from (0, 0) to (8, 8). You should know one quick formula for the cardinality of that set. However, counting a different way can lead to an interesting identity involving binomial coefficients. Notice that any path goes through exactly one of the points (0, 8),(1, 7),(2, 6), (3,5), (4,4), (5,3) ,(8, 0). Count the number of lattice paths that go through each of those 9 points - leave the expression in terms of binomial coefficients. Even more interesting is what you get if you generalize to a destination of (n, n), n 2 1. I am unsure of how to go about solving this problem in any way that makes sense, Was wondering if anyone here could help?

Consider the set of lattice paths from (0, 0) to (8, 8). You should know one quick formula for the cardinality of that set. However, counting a different way can lead to an interesting identity involving binomial coefficients. Notice that any path goes through exactly one of the points (0, 8),(1, 7),(2, 6), (3,5), (4,4), (5,3) ,(8, 0). Count the number of lattice paths that go through each of those 9 points - leave the expression in terms of binomial coefficients. Even more interesting is what you get if you generalize to a destination of (n, n), n 2 1. I am unsure of how to go about solving this problem in any way that makes sense, Was wondering if anyone here could help?

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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Question

Consider the set of lattice paths from (0, 0) to (8, 8).
You should know one quick formula for the
cardinality of that set. However, counting a different
way can lead to an interesting identity involving
binomial coefficients. Notice that any path goes
through exactly one of the points (0, 8),(1, 7),(2, 6),
(3,5), (4,4), (5,3) ,(8, 0). Count the number of lattice
paths that go through each of those 9 points - leave
the expression in terms of binomial coefficients.
Even more interesting is what you get if you
generalize to a destination of (n, n), n > 1.
I am unsure of how to go about solving this problem
in any way that makes sense, Was wondering if
anyone here could help?

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