First of all, I don't know how this method (in cryptography) is called in English. But the idea is that if we use as a cipher a vector with three coordinates, we can align these coordinates in 3! ways. That is: $$\left(\begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2 & \\ 3 & 2 & 1 \\ 1 & 3 & 2 \end{matrix} \right)$$
But I need a formal proof ( probably by mathematical induction) and I cannot find anywhere it online and cannot do it myself.
Maybe someone can share with the proof?
It is just the definition of the factorial. You can choose the first one in $n$ ways, the second in $n-1$ ways and so on. Yes, you can do it by induction. If you assume the result for $k$ items, for $k+1$ items you have $k+1$ ways to pick the first and $k!$ ways to order the rest, for a total of $(k+1)k!=(k+1)!$