We are asked to simply $$ H= \displaystyle \frac{1}{\ln 2}[\ln N! - \ln N_1! - \ln N_0!] $$ to $$rT\log_2\left(\frac{e}{r\Delta \tau }\right)$$ where $N_0 = (1-p)N,\, N_1 = pN, N = \displaystyle \frac{T}{\Delta\tau}, \, p = r\Delta\tau $, using Stirling's approximation $\ln x! \approx x(\ln x - 1)$.
So far I have, \begin{align} H &= \frac{1}{\ln 2}[N(\ln N - 1) - N_1(\ln N_1 - 1) - N_0(\ln N_0 - 1)]\\ &= \frac{1}{\ln 2}[N\ln N - N_1\ln N_1- N_0\ln N_0]\\ &= \frac{1}{\ln 2}[N\ln N - pN\ln pN- (1-p)N\ln (1-p)N]\\ &= \frac{1}{\ln 2}[N\ln N - pN\ln p- pN\ln N- (1-p)N\ln (1-p) - (1-p)N\ln N]\\ &= \frac{1}{\ln 2}[-pN\ln p- (1-p)N\ln (1-p)] \end{align} and that's where I'm stuck
I think the idea is that $p = r\Delta \tau$ is so small that we can approximate $\ln(1-p)$ by $\ln(1) = 0$, from that assumption it should be fairly straightforward.