2. Suppose n ≥ 1 is an integer. Consider an (n + 1) × (n + 1) grid of integer points; i.e. points of the form (a, b) where 0 ≤ a,b ≤n. A Binomial Path with 2n steps is a path from the point (0,0) to (n, n) formed by moving either 'right' (i.e. from (a, b) to (a +1, b)) or 'up' (i.e. from (a, b) to (a, b + 1)). (a) Draw all distinct Binomial Paths with 2n steps when n = 2. (b) Write down a correspondence that relates the Binomial Paths with 2n steps to strings of length 2n consisting of exactly n 1s and n Os. More precisely, let B be the set of Binomial Paths with 2n steps, and let Sn be the set of strings of length 2n consisting of exactly n 1s and n Os. Construct a bijection f: Bn → Sn. (You don't have to prove that it is a bijection.)
2. Suppose n ≥ 1 is an integer. Consider an (n + 1) × (n + 1) grid of integer points; i.e. points of the form (a, b) where 0 ≤ a,b ≤n. A Binomial Path with 2n steps is a path from the point (0,0) to (n, n) formed by moving either 'right' (i.e. from (a, b) to (a +1, b)) or 'up' (i.e. from (a, b) to (a, b + 1)). (a) Draw all distinct Binomial Paths with 2n steps when n = 2. (b) Write down a correspondence that relates the Binomial Paths with 2n steps to strings of length 2n consisting of exactly n 1s and n Os. More precisely, let B be the set of Binomial Paths with 2n steps, and let Sn be the set of strings of length 2n consisting of exactly n 1s and n Os. Construct a bijection f: Bn → Sn. (You don't have to prove that it is a bijection.)
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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2. Suppose n ≥ 1 is an integer. Consider an (n + 1) × (n + 1) grid of integer points; i.e.
points of the form (a, b) where 0 ≤ a,b ≤n. A Binomial Path with 2n steps is a
path from the point (0,0) to (n, n) formed by moving either 'right' (i.e. from (a, b) to
(a + 1, b)) or 'up' (i.e. from (a, b) to (a, b + 1)).
(a) Draw all distinct Binomial Paths with 2n steps when n = = 2.
(b)
Write down a correspondence that relates the Binomial Paths with 2n steps to
strings of length 2n consisting of exactly n 1s and n Os. More precisely, let B, be
the set of Binomial Paths with 2n steps, and let Sn be the set of strings of length
2n consisting of exactly n 1s and n 0s. Construct a bijection f: Bn → Sn. (You
don't have to prove that it is a bijection.)
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