You’re in a coding interview. The interviewer says: “A robot can only move right or down on an m × n grid. How many unique paths from the top-left to the bottom-right?” This is a classic 2D DP problem — but the sharper answer recognizes it’s a combinatorics problem solvable in O(1) space.

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The Problem

There is a robot on an m x n grid. It can only move right or down. Return the number of unique paths from the top-left to the bottom-right corner.

Example

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

Already comfortable with the solution? Practice it in a mock interview →

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What Interviewers Are Actually Testing

Skill	What Interviewers Observe
DP recurrence	Can you define dp[r][c] and its recurrence?
Space optimization	Can you reduce from 2D to 1D DP?
Combinatorics insight	Do you recognize this as C(m+n-2, m-1)?

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Step 1: The Key Insight

Every path uses exactly (m-1) down moves and (n-1) right moves. The number of unique paths = the number of ways to arrange these moves = C(m+n-2, m-1).

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Python Implementation

1D DP (space-optimized):

def uniquePaths(m: int, n: int) -> int:
    dp = [1] * n
    for _ in range(1, m):
        for c in range(1, n):
            dp[c] += dp[c-1]
    return dp[n-1]

Combinatorics (O(1) space):

from math import comb

def uniquePaths(m: int, n: int) -> int:
    return comb(m + n - 2, m - 1)

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Time & Space Complexity

Approach	Time	Space
1D DP	O(m × n)	O(n)
Combinatorics	O(min(m,n))	O(1)

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Common Interview Mistakes

1. Forgetting base cases. First row and column are all 1s — only one way to reach any edge cell.
2. Not knowing the combinatorics formula. Mentioning C(m+n-2, m-1) even if you implement DP is a strong signal.
3. Confusing m and n. Keep row/column indexing consistent.

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What a Strong Candidate Sounds Like

“The DP recurrence is dp[r][c] = dp[r-1][c] + dp[r][c-1]. I can space-optimize to O(n) with a 1D array. The mathematical insight is that every path is (m-1) downs and (n-1) rights interleaved — C(m+n-2, m-1) combinations.”

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Related Problems to Practice

• Climbing Stairs — 1D version of the same recurrence.
• Coin Change — DP with optimization choices.
• Maximum Subarray — DP pattern recognition.

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Practice This in a Mock Interview

Start a mock interview for Unique Paths on Intervu.

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Further reading:

• How to Prepare for a Coding Interview in 2026, the complete roadmap
• Why LeetCode alone isn’t enough, and what to practice instead
• Practice any LeetCode problem as a live mock interview
• The Grind 75 Pathway, a structured study plan with AI practice links
• Browse all walkthroughs on GitHub