Question: As stated in the title, how is the following simplification performed?
$$\frac{\left(\frac{n}{e}\right)^n}{\left(\frac{n}{2e}\right)^n}=2^n$$

This was shown by a student in this [Recitation video](http://www.youtube.com/watch?feature=player_embedded&v=P7frcB_-g4w) (29:05) from MIT OpenCourseware.

Answer: 
$$\frac{\left(\frac{n}{e}\right)^n}{\left(\frac{n}{2e}\right)^n} = \frac{n^n (2e)^n}{e^nn^n}$$
by $\frac{a/b}{c/d} = \frac{ad}{bc}$.

Clearly, the $n^n$ terms cancel, and $(2e)^n = 2^ne^n$. Then, the $e^n$ terms cancel.
$$\frac{\left(\frac{n}{e}\right)^n}{\left(\frac{n}{2e}\right)^n} =\frac{\left(\frac{n}{e}\right)^n}{\left(\frac{1}{2}\right)^n\left(\frac{n}{e}\right)^n} =\frac{1}{1/2^n}=2^n$$