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I do not think there is a formula that always works for degeneracy. But it can be calculated using simple math. For example, if there are three particles and each can be in one of the two directions, then the degeneracy is 2^3=8.
The formula you would use is w^n, where w is the number of possible states for the system, and n is the number of atoms. This equation shows us through the n exponent that degeneracy strongly depends on the number of atoms rather than the possible states.
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