I gave this a shot as well as since your value for E(T) → ∞ as T → ∞, while I would think the system should cap out at εN.

I get a different value for S(E), reasoning:

If E/ε is 1, there are N microstates, since 1 of N positions is at energy ε. If E/ε is 2, there are N(N-1) microstates. etc. etc, giving for E/ε = x that there are N!/(N-x)!

so S = ln [N!/(N-x)!] = ln(N!) - ln((N-x)!) = NlnN - (N-x)ln(N-x)

S(E) = N ln N - (N - E/ε) ln (N - E/ε)

Can you explain how you got your equation for the entropy?

Going on I get E(T) = ε(N - e^(ε/T - 1) )

This also looks wrong, as although E → ∞ as T → ∞, it also doesn't cap at exactly εN, and E → -∞ for T→ 0...

I'm expecting the answer to look something like: E(T) = εN(1 - e^(-ε/T))/2 which ranges from 0 to εN/2, which seems sensible.

EDIT: Nevermind, the answer was posted while I was writing this. I'd still like to know how you got your S(E) though.