How to use binomial theorem

Question

How to use binomial theorem，From (1) to get (2)?

$$\begin{align*}\left(1+\frac{1}{n}\right)^n\tag{1}\end{align*}$$

$$\begin{align*}t_n=1+1+\frac{1}{2!}\left(1-\frac{1}{n}\right)+\frac{1}{3!}\left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right)+\text{...}+\frac{1}{\text{n1}}\left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right)\text{...}\left(1-\frac{n-1}{n}\right).\tag{2}\end{align*}$$

I can get the following via using $C_n^k$ or $\left(\begin{array}{c} n \\ k \end{array}\right)$

$$\begin{align*}1+\frac{1}{n^5}+\frac{5}{n^4}+\frac{10}{n^3}+\frac{10}{n^2}+\frac{5}{n}\end{align*}$$

Answer 1

Hint: $$\binom{n}{i}\left(\frac{1}{n}\right)^{i}=\frac{n(n-1)\cdots(n-i+1)}{i!}\cdot\left(\frac{1}{n}\right)^{i}=\frac{1}{i!}\cdot\frac{n}{n}\cdot\frac{n-1}{n}\cdots\frac{1}{n}\\=\frac{1}{i!}\cdot1\cdot\left(1-\frac{1}{n}\right)\cdot\left(1-\frac{2}{n}\right)\cdots\frac{1}{n}$$

Answer 2

Hint:

Expand $C_n^k\cdot (\frac{1}{n})^k$.