Suppose we have a three-dimensional grid. An “up-right” path is considered to be one whose steps are of one of three forms: (x, y, z) → (x + 1, y, z) or (x, y, z) → (x, y + 1, z) or (x, y, z) → (x, y, z + 1).

a) How many such paths connect (0, 0, 0) with (4, 5, 6)?

b) How many such paths connect (0, 0, 0) with (4, 5, 6) via (2, 3, 5)?

c) Assume all up-right paths are equally likely, what is the probability that an up-right path from (0, 0, 0) to (4, 5, 6) passes through (2, 3, 5)?

*analogy with multinomial coefficient*