I'm starting a mini-course on Combinatorics and, although I can "see" the results, I'm having difficulties proving them.
For instance,
Being $\#A = n \in \mathbb{N}$, prove that $\#A^{k} = n^{k}$, for $k \in \mathbb{N}$.
I understand that $A^{k} = \{(x_1, \ldots, x_k) : x_i \in A\}$, and that we have $n \times n \ldots \times n$ ($k$ times) possible layouts for a sequence like $(x_1, \ldots, x_k)$, but I don't know what to use to prove the result.
Can you suggest some plan? If possible, could it be an advice that would give me traction to prove the following related results?
Thanks for the time of took to read my question. I highly appreciate it.
Update: Since this question is so straightforward (see comments), and was only presented with the intent of getting acquainted with the tools of the trade, I shall move on to another result I wish I can prove.
Prove that the number of elements in $\{(x_1, \ldots, x_k) : x_i \in A, i \in \{1, \ldots, k\}, x_i \neq x_j \Longleftrightarrow i \neq j, j \in \{1, \ldots, k\}\}$ is given by the formula $\frac{n!}{(n-k)!}$.
Thanks for your replies so far!