Right up and away

Suppose a particle, starting at the origin, moves in six positive integral steps, (i.e., each step is of integral length $$\ge 1$$), in the pattern right, up, right, up, right, up, such that the combined lengths of the steps is $$10$$, and such that each possible sequence of steps is equally likely to occur.

If $$D$$ is the expected (magnitude of the) distance between the origin and the particle after it has completed the six steps, find $$\lfloor 1000*D \rfloor.$$

Clarifications:

By "right" I mean in the positive $$x$$-direction, and by "up" I mean in the positive $$y$$-direction.

As an example, one possible path is $$2$$ right, $$2$$ up, $$1$$ right, $$3$$ up, $$1$$ right and $$1$$ up.

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