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 B₁ 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.)

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 B₁ 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.)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.7: Distinguishable Permutations And Combinations

Problem 30E

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