Prove $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^n = n!$ and $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^m = 0$ 
Prove
$$\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^n = n! \qquad\text{and}\qquad \sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^m = 0,$$
where $m \in \{0, 1, 2, \cdots, n - 1\}$.

To be honest, I tried to use some combinatorial ideas but was not able to see interconnections. Also, I had an idea to prove it using induction but failed to do that either.
Help will be much appreciated! I just got stuck and don't see any continuation of solution for this problem

Consider the exponential generating function
$$ f(x) = \sum_{m=0}^{\infty} \frac{A_m}{m!} x^m, \qquad A_m := \sum_{k=0}^{n} (-1)^{n-k}\binom{n}{k} k^m.  $$
Then it follows that
\begin{align*}
f(x)
&= \sum_{k=0}^{n} (-1)^{n-k}\binom{n}{k} \sum_{m=0}^{\infty} \frac{(kx)^m}{m!} \\
&= \sum_{k=0}^{n} (-1)^{n-k}\binom{n}{k} e^{kx} \\
&= (e^x - 1)^n.
\end{align*}
Writing $e^x - 1 = x g(x)$ for $g(x) = \sum_{j=0}^{\infty} \frac{x^j}{(j+1)!}$,
$$ f(x) = x^n g(x)^n . $$
In particular, the Maclaurin series for $f$ about $x = 0$ begins with $ f(x) = x^n + \cdots $, which translates to the desired fact:
$$ A_m
= \begin{cases}
0, &\text{if $m < n$}, \\
n!, &\text{if $m = n$}.
\end{cases} $$
In general, this computation shows that
$$ A_m = \sum_{\substack{j_1\geq1,\dots,j_n\geq1 \\ j_1+\cdots+j_n=m}} \binom{m}{j_1,\dots,j_n} \tag{1} $$
for all $m = 0, 1, 2, \dots $.

Alternatively, here is an extended version of @Trevor Gunn's answer: Let $[n] = \{1, \dots, n\}$. Then define
$$ X_i = \{ f \in [n]^{[m]} : i \notin \operatorname{range}(f) \} = ([n]\setminus\{i\})^{[m]} $$
for each $i = 1, 2, \dots, n$ and
$$ X_{\varnothing} = [n]^{[m]}, \qquad X_I = \bigcap_{i\in I} X_i, $$
for each non-empty $I \subseteq [n]$. Then $A_m$ can be written as
$$ A_m
= \sum_{k=0}^{n} (-1)^k \binom{n}{k} (n-k)^m
= \sum_{I\subseteq[n]} (-1)^{\left| I \right|} \left| X_I \right|. $$
Now by the inclusion-exclusion principle, it follows that
\begin{align*}
A_m
&= \left| \{ f \in [n]^{[m]} : f \notin X_1 \cup \cdots \cup X_n \} \right| \\
&= \text{[# of surjections from $[m]$ to $[n]$]}. \tag{2}
\end{align*}
Note that this formula also matches the formula $\text{(1)}$ in the first solution. Now the desired answer easily follows from $\text{(2)}$.
