a) In how many ways can a particle move in the xy-plane from the origin to the point (7, 4...

a) In how many ways can a particle move in the xy-plane from the origin to the point (7, 4) if the moves that are allowed are of the form:

(R): (x, y) →(x + 1, y); (U): (x, y) → (x, y + 1)?

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b) How many of the paths in part (a) do not use the path from (2, 2) to (3, 2) to (4, 2) to (4, 3) shown in Fig?

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c) Answer parts (a) and (b) if a third type of move

(D): (x, y) → (x + 1, y + 1)

 is also allowed.

Figure