Upper bound for sum of multinomial coefficients

Question

My question is the following: Is there any upper bound for the next sum?

$$ \sum_{\vert \alpha \vert=k}\frac{k!}{\alpha!} $$

where $k$ is a given positive interger and $\alpha\in \mathbb{N}^{n}$ such that $\alpha=(a_1, \cdots, \alpha_n)$, $\vert \alpha \vert=\sum_{i=1}^{n}\alpha_i$ and $\alpha!=\alpha_1!\alpha_2!\cdots \alpha_n!$

Answer 1

Your sum is $n^k$.

You know $$\left(\sum_{i=1}^n x_i\right)^k = \sum_{|\alpha|=k} \prod_{i=1}^n x_i^{a_i} \frac {k!}{\prod_{i=1}^n a_i!};$$ now evaluate at $x_1=x_2=\cdots=x_n=1.$