Feynman Checkerboard!

Probability Level 4

Richard P Feynman starts his journey from O=(0,0)O=(0,0) towards A=(10,6).A=(10,6). He can only go in the +x+x and +y+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 OO to AA in this way?


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

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