README.md

62. Unique Paths

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.

Solutions (Rust)

1. Dynamic Programming

impl Solution {
    pub fn unique_paths(m: i32, n: i32) -> i32 {
        let m = m as usize;
        let n = n as usize;
        let mut dp = vec![vec![0; n]; m];
        dp[m - 1][n - 1] = 1;

        for i in (0..m).rev() {
            for j in (0..n).rev() {
                if i < m - 1 {
                    dp[i][j] += dp[i + 1][j];
                }
                if j < n - 1 {
                    dp[i][j] += dp[i][j + 1];
                }
            }
        }

        dp[0][0]
    }
}

2. Mathematical

impl Solution {
    pub fn unique_paths(m: i32, n: i32) -> i32 {
        (1..(m as i64)).fold(1, |acc, x| acc * (n as i64 - 1 + x) / x) as i32
    }
}