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 \(\lfloor 1000*S \rfloor.\)

Comments:

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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