2017/18 AHSMC Part 1 Problem 7

Problem 7 of 16

The number of ways to walk from $(0,0)$ to $(20,2)$ by using only up and right unit steps and such that the walk never visits the lines $y = x$ and $y = x - 18$ , except at the beginning and end, is

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Solution

The first two steps must be to the right from $(0,0)$ to $(2,0)$ and the last two steps must be to the right from $(18,2)$ to $(20,2)$ . The number of ways of walking from $(2,0)$ to $(18,2)$ is $\dbinom{18}{2} = 153$ . We have to subtract one way that contains $(2,2)$ and the one which contains $(18,0)$ , hence the number of admissible ways is $151$ . The answer is (A).

University of Alberta

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