Indistinguishability in paramagnets

$$Z_N=Z_1^N$$

This reminds us of the Gibbs paradox. 

Next step:

$$Z_N=\frac{Z_1^N}{N!},$$

but since $$m=-\frac{dF}{dB}$$

it still doesn't change magnetization as the relation to \(Z_1\) still hasn't changed. So the ultimate solution would be

$$Z_N=\frac{(N+Z_1-1)!}{N!(Z_1-1)!},$$

and

$$Z_N=\frac{Z_1!}{N!(Z_1-N)!}.$$