How to simplify using stirling approximation?

[eprparadox]

Hey!

So we're deriving something in Daniel Schroeder's Introduction to Thermal Physics and it starts with this:

$$\Omega \left( N,q\right) =\dfrac {\left( N-1+q\right) !} {q!\left( N-1\right) !}$$

Both N and q are large numbers and q >> N.

The derivation is in the book, but I am always confused with when I can throw away terms.

For example, in this case intuitively, I would want to say that since q >> N and we have N - 1 + q, I would just keep the q and throw away the N - 1. I know this is wrong, but I don't know why.

In the book, they begin by throwing away the "- 1" term since both q and N are large numbers. I would have thrown away the N - 1. Why can't I do this and when can I/can't I throw away terms?

Thanks!

 

[Mentallic]

Because N is involved in the numerator's factorial. Since factorials grow very fast, then a larger factorial will contribute quite a lot, even though the added size isn't quite that large in comparison.
Example, if N=10 and q=1000, then q! is on the order of 10²⁵⁶⁷ and while (q+N)! is on the order of 10²⁵⁹⁷ and may not seem much larger, it's in fact 10³⁰ (a million trillion trillion) times larger.
$$q!=1\times2\times3\times...\times999\times1000 \approx 10^{2567}$$
$$N! = 1\times2\times...\times10 \approx 10^6$$
$$(q+N)! = 1\times2\times3\times...\times999\times1000\times 1001\times 1002 \times ... \times 1010$$
$$= q! \times 1001 \times 1002 \times... \times 1010 \approx q! \times 10^{30}$$
N! is only 10⁶ on its own, but when included with q in the factorial, it multiplies by 10³⁰. If you neglect a factor of this size, it'll definitely make a noticeable difference in your approximation.

 

[eprparadox]

Hey, got it. Thanks so much!