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

**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 \((x_1,y_1)\) to \((x_2,y_2)\) is \((x_2-x_1,y_2-y_1)\).

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

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