2023/24 AHSMC Part 1 Problem 10

Problem 10 of 16

An ant is at the point $(0,0)$ and every second it can move one unit either up or right. However, it can only move to the points $(x,y)$ where $x$ and $y$ are not both odd. The number of distinct paths the ant can follow to reach the point $(6,12)$ is

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Solution

When the ant is at even-even coordinates, it must advance either two steps right or two steps up to reach a new even-even point. Condensing each double step into one step, the problem becomes the number of unrestricted paths from $(0,0)$ to $(3,6)$ using only up and right moves. There are $9$ steps in total, so the number of paths is $\dbinom{9}{3} = 84$ . The answer is (D).

University of Alberta

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