Stirlings approx/CoinFlips/Gamma function

[thatguy14]

Homework Statement

Hi I actually have three questions that I am posting here, help in all of them would be greatly appreciated!

1) Prove that ln(n!) ≈ nln(n)-n+ln(2*pi*n)/2 for large n

2)Supposed you flip 1000 coins, what is the probability of getting exactly 500 heads

3) Show that n! = \int x^{n}exp(-x)dx where n is an integrer and the injtegral is from 0 to infinity. (The gamma function extends this definition of factorial to include non-integrers, z).

Homework Equations

Below

The Attempt at a Solution

1) I am pretty lost here. This is a third year thermodynamics course (not been heavily math based so far) so the solution shouldn't be something above that level. I don't even know where to start.

2) This one I thought would be really easy (and it should be). This is what I did.

I started off by ignoring the last term of stirlings approximation (this is what the textbook has and my prof did). the probability is

P_{N}(n) = \frac{N!}{(N-n)!n!}*2^{-N}
by using stirlings approximation, (first taking the natural log of both sides) and simplifying I got (and my professor)

P = exp(NlnN - (N-n)ln(N-n)-nln(n)-Nln(2))

but when I plug in 1000 for N and 500 for n I keep getting 1 which I don't think is right. What am I doing wrong?

3)I tried to do an integration by parts on the right side but it leads to some undefined things (i.e. 0*infinity) so I am lost yet again. I thought that was just a definition, how do I show that?

Thanks

 

[thatguy14]

Please ignore number two, it required that the terms left off to be added. Still need help with 1 and 3