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Questions on partition function

Questions on partition function from 'Fundamentals of STATISTICAL AND THERMAL PHYSICS'. See attached file for full problem description.


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Your auxiliary picture "4 help.jpg" seems to be answering the 2nd question, but I am not sure of the relevance of "1 help.jpg" to the 1st question, and so I offer a "start from basics" derivation of the relations requested in question 1.

Both the pdf file and its plain tex source are attached

bf Question 1 rm
The probability that a system in thermal equilibrium is in microstate $j$ is
p(j) propto e^{-beta E_j},
where $beta = 1/kT$. The ...