# This way and that

Discrete Mathematics Level 5

A particle in $$\mathbb{R}^{3}$$, starting at the origin, moves in six steps of positive integral length in the sequence east, north, up, east, north, up, where "east" means in the positive $$x$$-direction, "north" in the positive $$y$$-direction and "up" in the positive $$z$$-direction. The combined length of the six integral steps is $$10.$$

If the expected (magnitude of the) distance between the starting and finishing points of the particle is $$S,$$ then find $$\lfloor 1000*S \rfloor.$$

• For example, one possible path for the particle is $$2$$ units east, $$2$$ units north, $$1$$ unit up, $$1$$ unit east, $$3$$ units north and $$1$$ unit up.

• By "positive integral length" I mean that each step has a length $$\ge 1.$$

• Each possible path has an equal chance of being taken.

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