Verify that the quantity \((k / \mathcal{N}) \ln \Gamma\), where

\[
\Gamma(\mathcal{N}, U)=\sum_{\left\{n_{r}ight\}}^{\prime} W\left\{n_{r}ight\}
\]

is equal to the (mean) entropy of the given system. Show that this leads to essentially the same result for \(\ln \Gamma\) if we take, in the foregoing summation, only the largest term of the sum, namely the term \(W\left\{n_{r}^{*}ight\}\) that corresponds to the most probable distribution set.

[Surprised? Well, note the following example:

For all \(N\), the summation over the binomial coefficients \({ }^{N} C_{r}=N ! /[r !(N-r !)]\) gives

\[
\sum_{r=0}^{N}{ }^{N} C_{r}=2^{N}
\]

therefore,

\[
\begin{equation*}
\ln \left\{\sum_{r=0}^{N}{ }^{N} C_{r}ight\}=N \ln 2 \tag{a}
\end{equation*}
\]

Now, the largest term in this sum corresponds to \(r \simeq N / 2\); so, for large \(N\), the logarithm of the largest term is very nearly equal to

\[
\begin{align*}
& \ln \{N !\}-2 \ln \{(N / 2) !\} \\
\approx & N \ln N-2 \frac{N}{2} \ln \frac{N}{2}=N \ln 2, \tag{b}
\end{align*}
\]

which agrees with (a).]