leetcode: 62. Unique Paths

题目

Medium

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?

Note: m and n will be at most 100.

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

思路一

是一个排列组合题。

    def uniquePaths(self, m: int, n: int) -> int:
        total = m+n-2
        ans = 1
        for i in range(n-1):
            ans = ans * (total-i)
        for i in range(n-1):
            ans = ans//(i+1)
        return ans
        

思路二

用递归，m 行 n 列的路线数 = （m-1）行 n 列的路线数 + m 行 (n-1) 列的路线数

import collections
def uniquePaths(self, m: int, n: int) -> int:
    paths = collections.defaultdict(lambda:1)
        for i in range(2,m+1):
            for j in range(2,n+1):
                paths[str(i)+","+str(j)] = paths[str(i-1)+","+str(j)] + paths[str(i)+","+str(j-1)]
        return paths[str(m)+","+str(n)]