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


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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