Struggling to follow the transformation from $n \cdot e \left( n-1 \right)\left( \frac{n-1}{e}\right)^{n-1}$

Question

I'm going through the proof in Matousek's discrete maths book and I don't understand how he transforms this: $$n \cdot e \left( n-1 \right)\left( \frac{n-1}{e}\right)^{n-1}$$ to this: $$\left[en \left( \frac{n}{e} \right)^n \right] \cdot \left(\frac{n-1}{n} \right)^n e$$

Answer 1

We have $ \left( \frac{n-1}{e}\right)^{n-1}= \frac{e}{n-1}\left( \frac{n-1}{e}\right)^{n}$. The term becomes

$$n \cdot e \left( n-1 \right)\frac{e}{n-1}\left( \frac{n-1}{e}\right)^{n}$$

Cancelling $(n-1)$

$$n \cdot e \cdot e\left( \frac{n-1}{e}\right)^{n}$$

Multiplying the term by $\left( \frac{n}{n}\right)^{n}$. This does not change the value of the term since $\frac{n}{n}=1$

$$n \cdot e \cdot \left( \frac{n}{\color{blue}n}\right)^{n}\cdot e\left( \frac{n-1}{\color{blue}e}\right)^{n}$$

Exchanging the blue terms. Both are at the denominator and have an exponent $n$.

$$n \cdot e \cdot \left( \frac{n}{e}\right)^{n}\cdot e\left( \frac{n-1}{n}\right)^{n}$$