Maths N-step 1-d Random walk distribution

Question

I am trying to follow the attached (or if it's easier p.18-21 http://physics.bnu.edu.cn/application/faculty/tuzhanchun/biophys/L08.pdf on this).

I follow up to the bottom line on page 20 - the second logarithm expression and the Gaussian distribution on the top of page 21. Has the Gaussian expression been obtained from the logarithm expression?

I'm pretty stuck as to attempt what has even been done, what have we taken the log of and why?

Many thanks

Answer 1

I assume this is a demonstration of the normal approximation to the binomial distribution (see also the central limit theorem). I think they intend for you to apply the approximation $\log n! \approx n \log n - n + \frac{1}{2} \log(2 \pi n)$ for each factorial in $\frac{N!}{n_r!(N-n_r)!}$.