0062: Unique Paths

Problem Statement

There is a robot on an m x n grid. The robot is initially located at the top-left corner (i.e., grid[0][0]). The robot tries to move to the bottom-right corner (i.e., grid[m - 1][n - 1]). The robot can only move either down or right at any point in time.

Given the two integers m and n, return the number of possible unique paths that the robot can take to reach the bottom-right corner.

The test cases are generated so that the answer will be less than or equal to 2 * 109.

Example 1:

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

Example 2:

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 -> Down -> Down
2. Down -> Down -> Right
3. Down -> Right -> Down

Constraints:

  • 1 <= m, n <= 100

Code Solution

class Solution:
    def uniquePaths(self, m, n):
	    
        @cache
        def rec(rx, cx):
            if rx == m - 1 and cx == n - 1: return 0
            if rx == m - 1: return 1
            if cx == n - 1: return 1
            count = 0
            count = count + rec(rx + 1, cx)
            count = count + rec(rx, cx + 1)
            return count
        
        return rec(0, 0)