Why One Dimension Fails

Imagine counting paths from the top-left corner to the bottom-right corner of a grid. Can $dp[steps]$ = "number of paths after $k$ steps" work? No. Knowing you took 5 steps tells you nothing about WHERE you are.

You could be at cell $(3, 2)$ or $(1, 4)$. You need to track both your row and column: $dp[r][c]$ = "number of paths to reach cell $(r, c)$." Now the state captures everything you need: your exact position in the grid. When one number can't capture the full situation, you add another dimension. This is the core insight of multi-dimensional DP.