I want to distribute n labeled balls into m labeled boxes. I know one obtains the number by $m^n$. But I don't quite understand why. The underlying argument is always I have m choices for the first ball m choices for the second and so on. As an example lets take 3 balls labeled A,B,C and two boxes 1,2
Now if I make all combinations I can find much more than $2^{3}=8$ possibilities. As I understand the $m^{n}$ counting procedure neglects the permutations of the n balls and the permutations of the m boxes. For example taking the configuration $B_{1}(A,B,C)$ and $B_{2}()$, then why do I not count for example $B_{1}(C,B,A)$ and $B_{2}()$ as a distinct configurations. I was sketching all possibilities and this results in $m^{n}*m!*n!$. I mean this is clearly wrong based on google searches but why? I am really stuck with this problem