Start by clarifying the problem constraints and edge cases, then explain the dynamic programming approach where the number of paths to each cell is the sum of paths from the cell above and the cell to the left. Derive the recurrence relation and discuss how to optimize space, possibly mentioning the combinatorial solution.
Pro tip: Mention that the problem can be solved in O(min(m,n)) space by using a 1D array, and note that the combinatorial formula C(m+n-2, m-1) provides a direct mathematical solution, showing depth beyond basic DP.
Confirm that the grid is m x n, moves are only right or down, and the goal is to count unique paths from (0,0) to (m-1,n-1). Ask about edge cases like m=1 or n=1.
Let dp[i][j] be the number of unique paths to cell (i,j). Base case: dp[0][0] = 1. For first row and first column, there is only one path.
For other cells, dp[i][j] = dp[i-1][j] + dp[i][j-1] because you can arrive from above or from the left.
Use a 1D array of size n (or m) to compute row by row, updating dp[j] += dp[j-1] for each row, reducing space complexity to O(min(m,n)).
Mention the combinatorial solution: total moves = (m-1) downs + (n-1) rights, so paths = C(m+n-2, m-1). Time complexity O(m*n) for DP, O(min(m,n)) space; combinatorial O(min(m,n)) time.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.