Show that the entropy of an ideal gas in thermal equilibrium is given by the formula

\[
S=k \sum_{\varepsilon}\left[\left\langle n_{\varepsilon}+1ightangle \ln \left\langle n_{\varepsilon}+1ightangle-\left\langle n_{\varepsilon}ightangle \ln \left\langle n_{\varepsilon}ightangleight]
\]

in the case of bosons and by the formula

\[
S=k \sum_{\varepsilon}\left[-\left\langle 1-n_{\varepsilon}ightangle \ln \left\langle 1-n_{\varepsilon}ightangle-\left\langle n_{\varepsilon}ightangle \ln \left\langle n_{\varepsilon}ightangleight]
\]

in the case of fermions. Verify that these results are consistent with the general formula

\[
S=-k \sum_{\varepsilon}\left\{\sum_{n} p_{\varepsilon}(n) \ln p_{\varepsilon}(n)ight\}
\]

where \(p_{\varepsilon}(n)\) is the probability that there are exactly \(n\) particles in the energy state \(\varepsilon\).