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53. Unique Paths

mediumAsked at Reddit

Count the number of paths from top-left to bottom-right of an m x n grid (right/down only). Reddit uses this DP problem as a clean grid-DP warm-up before harder routing problems.

By Alex Chen, Founder, InterviewChamp.AI · Last verified

Source citations

Public interview reports confirming this problem appears in Reddit loops.

  • Glassdoor (2026-Q1)— Reddit phone screen, DP warm-up.

Problem

There is a robot on an m x n grid. The robot is initially located at the top-left corner. The robot tries to move to the bottom-right corner. 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.

Constraints

  • 1 <= m, n <= 100

Examples

Example 1

Input
m = 3, n = 7
Output
28

Example 2

Input
m = 3, n = 2
Output
3

Approaches

1. Recursive (no memo)

paths(i, j) = paths(i-1, j) + paths(i, j-1).

Time
O(2^(m+n))
Space
O(m+n)
function uniquePaths(m, n) {
  function dfs(i, j) {
    if (i === 0 && j === 0) return 1;
    if (i < 0 || j < 0) return 0;
    return dfs(i - 1, j) + dfs(i, j - 1);
  }
  return dfs(m - 1, n - 1);
}

Tradeoff: Exponential. TLE for medium m, n.

2. DP rolling 1D (optimal)

dp[j] = number of paths to (i, j). For each row, dp[j] += dp[j-1].

Time
O(m*n)
Space
O(n)
function uniquePaths(m, n) {
  const dp = new Array(n).fill(1);
  for (let i = 1; i < m; i++) {
    for (let j = 1; j < n; j++) {
      dp[j] += dp[j - 1];
    }
  }
  return dp[n - 1];
}

Tradeoff: O(n) memory by rolling rows. Math closed-form is C(m+n-2, m-1) but DP is interview-canonical.

Reddit-specific tips

Reddit interviewers expect the DP version with rolling memory. Bonus signal: mention the closed-form binomial C(m+n-2, m-1) and explain when you'd use one vs. the other.

Common mistakes

  • Allocating m*n 2D matrix when 1D rolling works.
  • Off-by-one on row/column initialization (fill with 1).
  • Not handling the m=1 or n=1 base case (answer is 1).

Follow-up questions

An interviewer at Reddit may pivot to one of these next:

  • Unique Paths II (LC 63) — with obstacles.
  • Minimum path sum (LC 64).
  • Dungeon game (LC 174).

Solve it now

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Output

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FAQ

Why does the closed-form work?

Every path has exactly m-1 downs and n-1 rights, in some order. Choose positions for the downs: C(m+n-2, m-1).

When would the 2D DP be needed?

When you need the path itself, or when transitions depend on the row direction.

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