Suppose I have $p$ P's, $q$ Q's, and $r$ R's. In how many different ways can I make a string of letters such that uses at most $p - 1$ P's, $r - 1$ R's, and uses all $q$ Q's?
I think it's $$\sum_{i = 0}^{p - 1}{\sum_{j = 0}^{r - 1}{\dfrac{(q + i + j)!}{q! \cdot i! \cdot j!}}}.$$ Can someone confirm or correct?
I think you made a typo. The equation should be $\sum_{i=0}^{p-1} \sum_{j=0}^{r-1} \frac{(q+i+j)!}{q!*i!*j!}$ and you are good to go.