Question

 Suppose someone e-mails you 10 different files you need to save one-by-one to your computer. Suppose that in total you have 6 different empty folders on your computer.

(i) How many ways can you save the files into the 6 folders if the temporal order in which you save the individual files does not matter.

(ii) How many ways can you save the files into the 6 folders if the temporal order in which you save the individual files matters.

Answer 1

i)

$N=\begin{pmatrix} n+ m-1\\ m \end{pmatrix}$

where m is number of files,

n is number of folders

$N=\begin{pmatrix} 6+ m-1\\ m \end{pmatrix}=\begin{pmatrix} 5+ m\\ m \end{pmatrix}=\frac{(5+m)!}{5!m!}$

ii)

$N=n^m=6^m$