Thermodynamics(math) derivation step

[iScience]

hi, i just need help with a step in a derivation in my thermodynamics book (indicated by the red arrow)



http://i.imgur.com/C8k3xzT.jpg

firstly, what's the point of a -(q+N) term if they're just going to cancel it out with a +q and +N?

basically i want to know how the terms inside the 'ln' can come out in front.

 

[TSny]

Note the mention of "Stirling's approximation". See

http://en.wikipedia.org/wiki/Stirling's_approximation