Small grid
Input
m = 3, n = 7
Output
28
There are C(3+7-2, 3-1) = C(8,2) = 28 distinct sequences of moves (down/right) to reach the end.
Full lesson preview
Count unique paths in a grid
Compute the number of unique paths from top-left to bottom-right in an m x n grid moving only right or down.
Python practice10 minDynamic ProgrammingBeginnerLast updated March 29, 2026
Problem statement
Given two integers m and n representing the number of rows and columns in a grid, count how many unique paths exist to move from the top-left corner (0,0) to the bottom-right corner (m-1,n-1) if you can only move either down or right at any point.
Write a function unique_paths(m, n) that returns the number of unique paths as an integer.
Task
Use combinatorics or DP to count the number of unique paths in a grid efficiently.
Examples
Input format
Two integers m and n (rows and columns).
Output format
An integer: number of unique paths from top-left to bottom-right.
Constraints
0 <= m, n <= 100; results may be large but fit in Python int. Aim for O(min(m,n)) or O(m*n) solutions.
Samples
Sample 1
Input
m = 1, n = 5
Output
1
Only one way when there's only one row: move right each time.