How do you count unique paths in a grid?

Question presented to candidate: "A robot starts in the top-left corner of an m × n grid and can only move right or down. How many different paths can it take to the bottom-right corner? After that, what changes if some cells are blocked? Please code it in Python."

What a strong answer should cover:

• The recurrence: a cell can only be entered from above or from the left, so paths[r][c] = paths[r-1][c] + paths[r][c-1], with the whole first row and first column equal to 1.
• O(m·n) time; space from O(m·n) down to O(n) by keeping a single row and updating it left to right.
• The closed form: every path is m−1 downs and n−1 rights in some order, so the answer is the binomial coefficient C(m+n−2, m−1), computable in O(min(m, n)) with exact integer steps.
• The obstacles variant: a blocked cell has 0 paths, and a blocked start or end makes the answer 0 — the formula no longer applies, the DP does.
• Python details: build 2D tables with a comprehension, not [[1] * n] * m (which shares one inner list), // for exact integer division, and math.comb for the binomial — with no overflow at any size.

Clarifying questions expected:

• Are the moves only right and down?
• How large can m and n be — does the answer fit in 32 or 64 bits?
• Are there obstacles, and can the start or end cell itself be blocked?

Code / implementation expected: Yes — a Python DP over the grid (one row of space is a plus); the binomial formula is a good follow-up.