What is the logic behind the combinations with repetitions formula This month, I'm taking combinatorics classes in my school, yesterday we learned about combinations with repetitions formula. Our teacher wrote it on the board, but she didn't really explained what is the logic behind this and why the reduction to combinations without repetition is working. I was curious to know how would you explain this in a more natural way other than just learning the formula as it is.
For those being unsure for which formula I mean here it is:
$$\bar{C}^{k}_{n} = C^{k}_{n+k-1}$$

Well, a $k$-repetition of $n$ is a word $x=x_1\ldots x_n$ of length $n$ over the alphabet ${\Bbb N}_0$ (natural numbers inluding $0$) such that $x_1+\ldots+x_n=k$.
The $2$-repetitions of $4$ are
$2000,1100,1010,1001,0200,0110,0101,0020,0011,0002$.
They count the number of ways to draw $2$ numbers from $1,2,3,4$ without looking at the order and with replacement (put them back). Here $2000$ accounts for the drawing of two $1$'s and $1100$ for the drawing of $1$ and $2$.
