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## Main Question or Discussion Point

How can I prove that, for [itex]N\gg n[/itex]

[itex]\frac{N!}{(N-n)!}\approx N^{n}[/itex]

I've tried doing

[itex]\frac{N!}{(N-n)!}=\exp\left(\ln\frac{N!}{(N-n)!}\right)=\exp\left(\ln N!-\ln\left(N-n\right)!\right)[/itex]

[itex]\underset{stirling}{\approx}\exp\left(N\ln N-N-\left(N-n\right)\ln\left(N-n\right)+N-n\right)[/itex]

[itex]=N^{N}+\left(N-n\right)^{n-N}+\exp-n[/itex]

But it doesn't look like I'm getting there

[itex]\frac{N!}{(N-n)!}\approx N^{n}[/itex]

I've tried doing

[itex]\frac{N!}{(N-n)!}=\exp\left(\ln\frac{N!}{(N-n)!}\right)=\exp\left(\ln N!-\ln\left(N-n\right)!\right)[/itex]

[itex]\underset{stirling}{\approx}\exp\left(N\ln N-N-\left(N-n\right)\ln\left(N-n\right)+N-n\right)[/itex]

[itex]=N^{N}+\left(N-n\right)^{n-N}+\exp-n[/itex]

But it doesn't look like I'm getting there