A robot is located at the top-left corner of a m x n grid (marked 'Start' in the diagram below).
The robot can only move either down or right at any point in time. The robot is trying to reach the bottom-right corner of the grid (marked 'Finish' in the diagram below).
How many possible unique paths are there?

這道題屬于非常簡單的一類dp問題，就不多說了。

class Solution {
    public int uniquePaths(int m, int n) {
        //state: f[i][j]:the number of unique paths to get to the position matrix[i][j]
        int[][] f = new int[m][n];
        //initialization:
        for (int i = 0; i < m; i++){
            f[i][0] = 1;
        }
        for (int i = 0; i < n; i++){
            f[0][i] = 1;
        }
        //function:
        for (int i = 1; i < m; i++){
            for (int j = 1; j < n; j++){
                f[i][j] = f[i - 1][j] + f[i][j - 1];
            }
        }
        //answer
        return f[m - 1][n - 1];
    }
}