Unique Paths

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?

Above is a 3 x 7 grid. How many possible unique paths are there?

Note: m and n will be at most 100.

方法一：相当于求C(m+n,m)，可以用递推公式C(x,y) = C(x-1,y-1) + C(x-1,y)

方法二：用动态规划，每次只能向右向下，所以(n,m)处的方法是走到(n-1,m)向右一步或者C(n,m-1)向下一步。

所以F(n,m) = F(n-1,m) + F(n,m-1)。代码如下：

class Solution {
public:
    int uniquePaths(int m, int n) {
        // Start typing your C/C++ solution below
        // DO NOT write int main() function
        int a[100][100];
        for(int i = 0;i < n;i++)
            a[i][0] = 1;
        for(int j = 0;j < m;j++)
            a[0][j] = 1;
            
        for(int i = 1;i < n;i++)
        {
            for(int j = 1;j < m;j++)
            {
                a[i][j] = a[i-1][j] + a[i][j-1];
            }
        }
        return a[n-1][m-1];
    }
};

8 milli secs