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. You are traveling on a grid from (2, 2) to (18, 18). The only steps you can make are either to increase the first number by 1, or to increase the second number by 1, which you can think of as unit steps that either go up or to the right. (a) Identify a bijection between the paths from (2, 2) to (18, 18) and bit sequences with a certain number of ones. Hint: there are two types of steps, so let one of these be a zero, and the other a one. (b) How many paths are there from (2, 2) to (18, 18)? (c) How many of these paths do not pass through (10, 10)? (d) How many of these paths do not pass through (7,7) or (10, 10)? (Note: this can also be phrased as paths that avoid both (7,7) and (10, 10), using De Morgan's Law to negate the or notice this is a time where "and" does not mean multiply in counting).

. You are traveling on a grid from (2, 2) to (18, 18). The only steps you can make are either to increase the first number by 1, or to increase the second number by 1, which you can think of as unit steps that either go up or to the right. (a) Identify a bijection between the paths from (2, 2) to (18, 18) and bit sequences with a certain number of ones. Hint: there are two types of steps, so let one of these be a zero, and the other a one. (b) How many paths are there from (2, 2) to (18, 18)? (c) How many of these paths do not pass through (10, 10)? (d) How many of these paths do not pass through (7,7) or (10, 10)? (Note: this can also be phrased as paths that avoid both (7,7) and (10, 10), using De Morgan's Law to negate the or notice this is a time where "and" does not mean multiply in counting).

Advanced Engineering Mathematics
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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. You are traveling on a grid from (2, 2) to (18, 18). The only steps you can make are either to increase the first number by 1, or to increase the second number by 1, which you can think of as unit steps that either go up or to the right. (a) Identify a bijection between the paths from (2, 2) to (18, 18) and bit sequences with a certain number of ones. Hint: there are two types of steps, so let one of these be a zero, and the other a one. (b) How many paths are there from (2, 2) to (18, 18)? (c) How many of these paths do not pass through (10, 10)? (d) How many of these paths do not pass through (7,7) or (10, 10)? (Note: this can also be phrased as paths that avoid both (7,7) and (10, 10), using De Morgan's Law to negate the or notice this is a time where "and" does not mean multiply in counting).
Transcribed Image Text:. You are traveling on a grid from (2, 2) to (18, 18). The only steps you can make are either to increase the first number by 1, or to increase the second number by 1, which you can think of as unit steps that either go up or to the right. (a) Identify a bijection between the paths from (2, 2) to (18, 18) and bit sequences with a certain number of ones. Hint: there are two types of steps, so let one of these be a zero, and the other a one. (b) How many paths are there from (2, 2) to (18, 18)? (c) How many of these paths do not pass through (10, 10)? (d) How many of these paths do not pass through (7,7) or (10, 10)? (Note: this can also be phrased as paths that avoid both (7,7) and (10, 10), using De Morgan's Law to negate the or notice this is a time where "and" does not mean multiply in counting).
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