Next, compute the number of paths that pass through the forbidden square at (3,3). To do this, split the path into two parts: from (1,1) to (3,3), and from (3,3) to (5,5).

February 22, 2026

Next, compute the number of paths that pass through the forbidden square at (3,3). To do this, split the path into two parts: from (1,1) to (3,3), and from (3,3) to (5,5).

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Question: A robotics engineer is programming a robot to navigate a grid that is 5 rows by 5 columns. If the robot starts at the bottom-left corner and must move only right or up to reach the top-right corner, how many distinct paths avoid the center square at (3,3)?
Solution: First, compute the total number of paths from the bottom-left to the top-right without restrictions. The robot must make 5 right (R) and 5 up (U) moves, so:
\binom{10}{5} = 252
From (1,1) to (3,3): 2 R and 2 U moves → $\binom{4}{2} = 6$
From (3,3) to (5,5): 2 R and 2 U moves → $\binom{4}{2} = 6$
So, number of paths through (3,3) is:

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