(a) Assuming that the total number of microstates accessible to a given statistical system is \(\Omega\), show that the entropy of the system, as given by equation (3.3.13), is maximum when all \(\Omega\) states are equally likely to occur.

(b) If, on the other hand, we have an ensemble of systems sharing energy (with mean value \(\bar{E}\) ), then show that the entropy, as given by the same formal expression, is maximum when \(P_{r} \propto \exp \left(-\beta E_{r}\right), \beta\) being a constant to be determined by the given value of \(\bar{E}\).

(c) Further, if we have an ensemble of systems sharing energy (with mean value \(\bar{E}\) ) and also sharing particles (with mean value \(\bar{N}\) ), then show that the entropy, given by a similar expression, is maximum when \(P_{r, s} \propto \exp \left(-\alpha N_{r}-\beta E_{s}\right), \alpha\) and \(\beta\) being constants to be determined by the given values of \(\bar{N}\) and \(\bar{E}\).