An ant moves one unit per second over the points 0, 1, and 2 in the following figure, starting from point 0.

![](https://cdn.mathpix.com/cropped/2024_05_01_98fe4f82538e82dc6d3ag-2.jpg?height=105&width=232&top_left_y=450&top_left_x=1089)

(a) What are the possible paths the ant can take in 3 seconds?

(b) How many possible paths can the ant take in 10 seconds?
(a) Up to three seconds, we have two possible paths: $0-1-0-1$ or $0-1-2-1$.

(b) Observe that when the ant is at points 0 and 2, it only has one possibility for the next second, which is to go to 1. When it is at 1, it has two possibilities for the next second, which is to go to 0 or 2. Thus, at odd seconds, the ant is always at 1, while at even seconds, it is at 0 or 2. Therefore, the number of possible paths after 10 seconds is

$$
1 \times 2 \times 1 \times 2 \times 1 \times 2 \times 1 \times 2 \times 1 \times 2=32
$$