Lattice Paths

How many {(1,1),(1,1),(2,0)}\{(1,1), (1,-1), (2,0)\}-lattice paths are there from the point (0,0)(0,0) to the line x=8?x = 8?

Details and assumptions

A lattice path is a path in the Cartesian plane between points with integer coordinates.

A step in a lattice path is a single move from one point with integer coordinates to another.

The size of the step from (x1,y1)(x_1,y_1) to (x2,y2)(x_2,y_2) is (x2x1,y2y1)(x_2-x_1,y_2-y_1).

For a set S={(xi,yi)}i=1kS = \{(x_i,y_i)\}_{i=1}^{k}, an SS-lattice path is a lattice path where every step has size which is a member of SS.

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