PPractick

The problem

You're building the same warehouse robot navigation simulator as before, but this time there are no obstacles — the warehouse floor is completely clear. A robot starts at the top-left corner of an m x n grid and needs to reach the bottom-right corner, moving only right or down at each step. Your task: compute the total number of distinct paths the robot can take from start to destination.

Example: a 3x7 grid → there are 28 unique ways to get from top-left to bottom-right moving only right or down.

Function signature

int uniquePaths(int m, int n);

Test cases

Input m	Input n	Output	Explanation
3	7	28	Standard grid, 6 rights + 2 downs arranged in any order
3	2	3	3 paths: RD-D, D-RD, DD-R style arrangements
1	1	1	Start = end, exactly one trivial "path" (do nothing)
1	5	1	Single row, only one path — go straight right
5	1	1	Single column, only one path — go straight down
7	3	28	Same as (3,7) — problem is symmetric in m and n

(Confirm: m, n >= 1; grid has no obstacles in this version.)

How interviewers phrase it

• "You're at the top-left corner of an m x n grid and want to reach the bottom-right, moving only right or down. How many unique paths are there?"
• This is the classic LeetCode 62 (Unique Paths) — often the interviewer's opening question before adding obstacles as a follow-up (Unique Paths II).
• Sometimes phrased combinatorially: "how many distinct sequences of moves get you from A to B" — a strong hint toward the combinatorics angle.

Clarifying questions to ask

1. Are the only allowed moves right and down — no diagonal, no up/left?
2. Are m and n guaranteed to be at least 1? What's the expected upper bound on m and n — does it hint at whether a DP or a closed-form combinatorial solution is expected?
3. Should the result fit in a standard int, or could it be large enough to need long long / modulo arithmetic for very large grids?
4. Is this a warm-up for a follow-up involving obstacles, or is a pure combinatorial answer acceptable and preferred?

Follow-ups to expect

1. "Can you solve this without DP, using combinatorics?"
2. "What if obstacles are introduced?"
3. "Can you compute the combinatorial answer without overflow for large m, n?"
4. "Can you reduce DP space to O(min(m,n))?"
5. "What if diagonal moves are also allowed?"
6. "Can you do this recursively with memoization instead of iterative DP or combinatorics?"

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