Rubber Band Model for 1D Chain of N Links

[KFC]

In the text, there is a rubber band model for 1D chain of N links. Assume each link of the rubber band is align horizontally either to the right or to the left. In the text, define $$N_+$$ as the number of links directed to the right and $$N_-$$ represents the number of links directed to the left. I think the total configuration of the system should be

$$\Omega = \frac{N!}{N_+!N_-!}$$

but it is

$$\Omega = \frac{2N!}{N_+!N_-!}$$

I wonder where is the "2" come from?

 

[Valerie Park]

The "2" comes from the fact that each link can be directed either to the right or to the left, meaning there are two possible configurations for each link. Therefore, when calculating the total number of configurations for the system, we need to take into account the two possibilities for each link, resulting in a factor of 2.

 

[charng]

I would like to clarify that the "2" in the formula represents the fact that each link in the chain has two possible orientations - either to the right or to the left. Therefore, when calculating the total number of configurations for a 1D chain of N links, we need to account for this possibility by multiplying by 2. This is because in a 1D system, the links can only be arranged in two ways - either all to the right or all to the left. Therefore, the correct formula for the total configuration of the system should be \frac{2N!}{N_+!N_-!}. This takes into account the two possible orientations for each link in the chain.