leetcode 62. Unique Paths （组合）

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

Example 1:

Input: m = 3, n = 2
Output: 3
Explanation:
From the top-left corner, there are a total of 3 ways to reach the bottom-right corner:

1. Right -> Right -> Down
2. Right -> Down -> Right
3. Down -> Right -> Right
   Example 2:

Input: m = 7, n = 3
Output: 28

Constraints:

1 <= m, n <= 100
It’s guaranteed that the answer will be less than or equal to 2 * 10 ^ 9.

• 把每种走法看成一个序列，每次可以选择右或者下
• 可以直到序列长度位 $m+n-2$
• 所以走法的总数就是这个序列用 $m-1$ 个下以及 $n-1$ 个右填充的方法数
• 可知答案为 $C_{n+m-2}^{m-1}$
• 防止溢出，展开计算：$m\times(m+1)\times\cdots\times (m+n-2)/(n-1)!$

class Solution {
public:
    int uniquePaths(int m, int n) {
        double res=1;
        for(int i=1;i<=n-1;i++)
            res=res*(m+i-1)/i;
        return (int)res;
    }
};