Using the approximate expression, see Fowler and Guggenheim (1940), [ g_{N}left(N_{1}, N_{12} ight) simeq frac{left(frac{1}{2} q N ight) !}{N_{11} ! N_{22} !left[left(frac{1}{2} N_{12} ight) !

Using the approximate expression, see Fowler and Guggenheim (1940),

\[
g_{N}\left(N_{1}, N_{12}\right) \simeq \frac{\left(\frac{1}{2} q N\right) !}{N_{11} ! N_{22} !\left[\left(\frac{1}{2} N_{12}\right) !\right]^{2}}\left(\frac{N_{1} ! N_{2} !}{N !}\right)^{q-1}
\]

for evaluating the partition function of an Ising lattice, show that one is led to the same results as the ones following from the Bethe approximation.