In how many ways can $r$ elements be chosen from $n$ elements with repetition, such that every element is chosen at least once? Thank you for your comments -- hopefully a clarifying example --
In choosing all combinations of 4 elements from (A,B,C), (e.g., AAAA,BBBB,ACAB,BBBC, etc.) how many of these combinations include A, B and C? Using a computer, I came up with 36 for this example. There are 81 combinations, and I found only 45 combinations that include 2 or less of the three elements. 
That is, where element order is considered and repetition allowed. I can't seem to get my head around it. What I came up with was n choose r - (n choose n-1) * (n-1 choose r) but the latter overlap and I was not sure how to calculate the overlaps.
Can anyone please help?
Thanks in advance.

If $r < n$, then the answer is $0$.
For $r \ge n$, the answer is $n! \times n^{r-n}$
