$$p_i = \frac{ \exp\left(-\frac{\epsilon _i}{k_BT} \right)}{Z} $$ $$ Z= \sum_{i} \exp\left(-\frac{\epsilon _i}{k_BT} \right)$$
A) Is $p_i$ the probability of the system having an energy equal to $\epsilon_i$? (Probability to be in any of the many microstates that have energy $\epsilon_i$).
B) Or is $p_i$ the probability of the system being in one particular microstate which happens to have energy $\epsilon_i$? (This microstate is not the only microstate with the same energy).
If A) is correct then: $$ Z= \sum_{\epsilon_i} \exp\left(-\frac{\epsilon _i}{k_BT} \right)$$
If B) is correct then: $$ Z= \sum_{\epsilon_i} \Omega_i\exp\left(-\frac{\epsilon _i}{k_BT} \right),$$ where $\Omega_i$ is the multiplicity of the macrostate of energy $\epsilon_i$.
From the derivation of the Boltzmann distribution I am inclined to understand it as B). But I have never seen the multiplicity in the partition function.
What is the correct interpretation of the Boltzmann distribution?