Question about combination of a set of numbers

Question

Suppose $x$ is a set of integers as below: $$x = (-1,0,1)$$ How many different sets of length 4 can be made by x such as: $$\begin{bmatrix} -1\ -1\ -1\ -1 \end{bmatrix}$$ $$\begin{bmatrix} -1\ 0\ -1\ -1 \end{bmatrix}$$ $$\begin{bmatrix} -1\ 0\ 1\ -1 \end{bmatrix}$$ and so on. In general, x is a set of length $k$, and it is aimed to create all possible different set of vectors $V$ of length $n$, where n > k. Note that for any $v_i$ and $v_j$ $\in$ $V$ the following holds: $$v_i\ -\ v_j\ \ne\ [0]$$ where $[0]$ is a vector of zeros of length $n$.

Answer 1

For a set $x=\{x_1, x_2, ..., x_k\}$ of length $k$, we want to find the set $V$ of vectors $v_l \in x^N$ (here $x^N$ means the power set of the set $x$, the same as $\mathbb{R}^N$). We now want to know how large $V$ is.

This can be modelled as urn problem drawing with putting back in consideration of the order. For the first entry of the vector of length $N$ you have $k$ different choices from the set $x$ (Note that, as $x$ is a set, there are no two identical elements in the set). As the order counts as well, you have these choices for every entry, which gives you $N^k$ different choices, i.e. $|V| = N^k$.