Paths dodging a closed intersection

Moving only right or up on the integer grid, how many lattice paths go from $(0,0)$ to $(5,4)$ without passing through the point $(2,2)$ ?

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Count all paths, then subtract the paths that pass through $(2,2)$ .
Paths through $(2,2)$ factor as $\binom{4}{2}\cdot\binom{5}{2}$ ; total is $\binom{9}{4}$ .

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