Multiplicity for 3 Dimensional problem

[Krazer101]

Homework Statement

I am having trouble finding an expression for W (multiplicity) for a chain that can move in all possible directions (3 dimensions)

Homework Equations

Multiplicity is the number of possible states over total states.

The Attempt at a Solution

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I understand for a chain that can move in 1 dimension (left or right), the multiplicity is N!/(n_r!(N-n_r)!). N is the number of monomers the chain is made from and n_r is the number of links pointing right and n_l = N - n_r, is the number of links pointing left. I was wondering how to find the multiplicity when the chain can movie in 3 dimensions (6 total directions)?

 

[diazona]

If I understand the problem correctly, think about this: the multiplicity for your 1D chain is the coefficient of $(x_{r})^{n_r}(x_{l})^{n_l}$ in the expansion of
$$(x_r + x_l)^N$$
Does that suggest anything to you? Any way to generalize this from 2 directions to 6?

 

[Krazer101]

Is it possible to raise the expansion to 3N instead of N to illustrate the other possible directions?

 

[diazona]

Well, remember what N represents: the number of links in the chain. If you did that, you'd be getting an expression for a chain with triple the length.

What in $(x_r + x_l)^N$ corresponds to the number of directions?

 

[Krazer101]

Oh I see, thank you