Feynman Checkerboard!

Discrete Mathematics Level 4

Richard P Feynman starts his journey from \(O=(0,0)\) towards \(A=(10,6).\) He can only go in the \(+x\) and \(+y\) directions along the lattice, and he must turn exactly 7 times. An example path is shown above, with the yellow dots indicating turns.

How many ways can Feynman travel from \(O\) to \(A\) in this way?

Bonus: Consider the general case, traveling from \((0,0)\) to \((p,q)\) with exactly \(r\) turns.

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