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How do you count unique paths in a grid?

Q: How do you count unique paths in a grid?

**Intuition.** A robot moving only **right** or **down** reaches a cell from exactly two places: the cell above and the cell to its left. So the number of paths to any cell is the **sum** of those two — and the edges (top row, left column) have exactly one path. Fill the grid and read the bottom-right corner. paths(cell) = paths(up) + paths(left) 1 <rect x="170" y="35" width="40" height="34

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Explain the grid paths DP.

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Rolling 1-D DP

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def unique_paths(m, n):
    dp = [1] * n                 # top row: one way to each cell
    for _ in range(1, m):
        for j in range(1, n):
            dp[j] += dp[j - 1]   # from above (dp[j]) + from the left (dp[j-1])
    return dp[n - 1]


# --- demo ---
print(unique_paths(3, 7))   # 28
print(unique_paths(3, 2))   # 3

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