A coin travels along a path that starts in an unshaded square in the top row of the figure, that uses only diagonal moves, and that ends in an unshaded square in the bottom row. A diagonal move takes the coin either one square down and one square left, or one square down and one square right. How many different paths from the top row to the bottom row are possible?
(A) 16
(B) 20
(C) 32
(D) 24
(E) 28

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-034.jpg?height=352&width=350&top_left_y=2166&top_left_x=1343)
We determine the number of ways to get to each unshaded square in the grid obeying the given rules.

In the first row, there is 1 way to get to each unshaded square: by starting at that square. In each row below the first, the number of ways to get to an unshaded square equals the sum of the number of ways to get to each of the unshaded squares diagonally up and to the left and up and to the right from the given square. This is because any path passing through any unshaded square needs to come from exactly one of these unshaded squares in the row above. In the second row, there are 2 ways to get to each unshaded square: 1 way from each of two squares in the row above.

In the third row, there are 2 ways to get to each of the outside unshaded squares and 4 ways to get to the middle unshaded square.

Continuing in this way, we obtain the following number of ways to get to each unshaded square:

| 1 |  | 1 |  | 1 |
| :--- | :--- | :--- | :--- | :--- |
|  | 2 |  | 2 |  |
| 2 |  | 4 |  | 2 |
|  | 6 |  | 6 |  |
| 6 |  | 12 |  | 6 |

Since there are 6, 12 and 6 ways to get to the unshaded squares in the bottom row, then there are $6+12+6=24$ paths through the grid that obey the given rules.

ANSWER: (D)