Starting with the 2, the number 2005 can be formed by moving either horizontally, vertically, or diagonally from square to square in the grid. How many different paths can be followed to form 2005?
(A) 96
(B) 72
(C) 80
(D) 64
(E) 88

| 5 | 5 | 5 | 5 | 5 |
| :--- | :--- | :--- | :--- | :--- |
| 5 | 0 | 0 | 0 | 5 |
| 5 | 0 | 2 | 0 | 5 |
| 5 | 0 | 0 | 0 | 5 |
| 5 | 5 | 5 | 5 | 5 |
From the centre 2 , there are 8 possible 0 s to which we can move initially: 4 "side" 0 s

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-185.jpg?height=249&width=260&top_left_y=1727&top_left_x=976)

and 4 "corner" 0s.

|  |  |  |  |  |
| :--- | :--- | :--- | :--- | :--- |
|  | 0 |  | 0 |  |
|  |  | 2 |  |  |
|  | 0 |  | 0 |  |
|  |  |  |  |  |

From a side 0 , there are 4 possible 0 s to which we can move: 2 side 0 s and 2 corner $0 \mathrm{~s}$

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-186.jpg?height=255&width=260&top_left_y=257&top_left_x=976)

From a corner 0 , there are 2 possible 0 s to which we can move: 2 side 0 s.

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-186.jpg?height=246&width=263&top_left_y=690&top_left_x=974)

From a side 0 , there are 3 possible 5 s to which we can move.

|  |  |  |  |  |
| :--- | :--- | :--- | :--- | :--- |
| 5 |  |  |  |  |
| 5 | 0 |  |  |  |
| 5 |  |  |  |  |
|  |  |  |  |  |

From a corner 0 , there are 5 possible 5 s to which we can move.

| 5 | 5 | 5 |  |  |
| :--- | :--- | :--- | :--- | :--- |
| 5 | 0 |  |  |  |
| 5 |  |  |  |  |
|  |  |  |  |  |
|  |  |  |  |  |

So the three possible combinations of 0s are "side-side", "side-corner" and "corner-side".

For the combination "side-side", there is a total of $4 \times 2 \times 3=24$ paths (because there are initially 4 side 0 s that can be moved to, and from each of these, there are 2 side 0 s that can be moved to, and from each of those, there are 35 s that can be moved to).

For the combination "side-corner", there is a total of $4 \times 2 \times 5=40$ paths.

For the combination "corner-side", there is a total of $4 \times 2 \times 3=24$ paths.

Therefore, in total there are $24+40+24=88$ paths that can be followed to form 2005 .

ANSWER: (E)