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

## Solution

from typing import List

class Solution:
def uniquePaths(self, m: int, n: int) -> int:
if m <= 1 or n <= 1:
return 1

path = [[1 for i in range(m)] for j in range(n)]

for i in range(n-2, -1, -1):
for j in range(m-2, -1, -1):
path[i][j] = path[i+1][j] + path[i][j+1]
return path[0][0]


## Test Cases

test = Solution()
assert test.uniquePaths(3, 2) == 3
assert test.uniquePaths(7, 3) == 28
print('All Passed!')


## Big O Analysis

Space Complexity: O(NM)

Time Complexity: O(NM)