Entropy of a Fermi dirac ideal gas

[Dassinia]

Hello

Homework Statement

From the expression of the partition function of a fermi dirac ideal gas
ln(Z)=αN + ∑ ln(1+exp(-α-βE_r))
show that
S= k ∑ [ <n_r>ln(<n_r>)+(1-<n_r>)ln(1-<n_r>)

Homework Equations

S=k( lnZ+β<E>)
<nr>=-1/β ∂ln(Z)/∂E_r
<E>=-∂ln(Z)/∂β

The Attempt at a Solution

I tried to start with
S=k( lnZ+β<E>)
But I don't know how we get to introduce the nr in that ?

Edit:
S=-dF/dT with F=-kTln(Z)
I wrote ln(Z) in terms of n_r
ln(Z)=α ∑<n_r>-∑ln(<n_r>)
But I don't get to the result
I get at the end
S= kα∑<n_r>-k∑ln(<n_r>)+kα∑(1-<n_r>)

Thanks

 

[rude man]

I suggest moving this to the advanced physics forum.