62. Unique Paths （走棋盘多少种不同的走法 动态规划）

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.

r如果用公式的话 就是 C(X+Y 选X) 

class Solution {
    public int uniquePaths(int m, int n) {
        int[][] dp = new int[m][n];

        for(int i =0;i<m;i++){
            for (int j = 0;j < n;j++){
                if(i==0&&j==0)
                    dp[i][j]=1;
                else if(i==0)
                    dp[i][j] = dp[i][j-1];
                else if(j==0)
                    dp[i][j] = dp[i-1][j];
                else
                 dp[i][j] = dp[i-1][j] + dp[i][j-1];
            }
        }
        return dp[m-1][n-1];
    }
}

class Solution {
    public int uniquePaths(int m, int n) {
        int[][] dp = new int[m][n];
        for(int i = 0;i<m;i++)
            dp[i][0] = 1;
        for(int i = 0;i<n;i++)
            dp[0][i] = 1;
        for(int i =1;i<m;i++)
            for (int j = 1;j < n;j++)
                dp[i][j] = dp[i-1][j] + dp[i][j-1];
        return dp[m-1][n-1];
    }
}