I'd like to know if there is a general formula or technique for this type of problems, which I describe generally as follows :
Say you have $n$ different colors and that you have $r_1$ balls of the first color, $r_2$ balls of the second color,..., $r_n$ balls of the $n$-th color. How many different sequences of $p$ balls, with $p < \sum_{k=1}^n r_k$ , are there?
It seems to me that it is not easy to calculate the number of different sequences.
In the case $p = \sum_{k=1}^n r_k$ it becomes a permutation with repetition problem because the total number is $\frac{p!}{r_1!r_2! \dots r_n!} $.
However, with $p < \sum_{k=1}^n r_k$ is trickier . How can it be done?
