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📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 \binom{13}{4} - \binom{5}{4} = 715 - 5 = 710

📌 Thus, the number of ways to select such a sequence is $\boxed{710}$.

📌 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

📌 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).

📌 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 → $\binom{4}{2} = 6$

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