# Question

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?

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

It is $${m+n-2\choose m-1}$$, which can be reduced to $$\frac{n\times \dots \times (m+n-2)}{1\times \dots \times (m-1)}$$.

class Solution:
def uniquePaths(self, m, n):
"""
:type m: int
:type n: int
:rtype: int
"""
if m < n:
m, n = n, m

numerator = 1
denominator = 1
for i in range(n, m + n - 1):
numerator *= i
for i in range(1, m):
denominator *= i

return numerator // denominator