From (3,3) to (5,5): 2 R and 2 U moves → $\binom{4}{2} = 6$
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- \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 → $\binom{4}{2} = 6$
- So, number of paths through (3,3) is:
- Thus, the number of valid paths avoiding (3,3) is:
- - 36 = \boxed{216}
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