Variation of distributing $k$ balls into $n$ distinguishable boxes I am interested in understanding a variation problem of distributing balls into boxes. It seems to be not any of the individual case mentioned in twelvefold way classification.
The problem is described as follows: 
In total there are $K$ balls, there are $I$ distinguishable groups by color, and in each color group the balls are indistinguishable with number to be $n_i$ where $i = 1, 2, \ldots, I$. Meanwhile, we have $N$ distinguishable boxes where $N$ is always bigger than any $n_i$. Therefore, how many ways are there to distribute these $K$ balls into $N$ boxes, when no box can contain more than one ball from same indistinguishable color group and no empty boxes?
Any suggestion will be appreciated.

I got $$P = \prod_{i=1}^{I} {N\choose n_i} - \prod_{i=1}^{I} {{N-1}\choose n_i}$$
The first part is obviously the combinations of "no box contains more than one ball of the color".
The second part is "no box contains more than one ball of the color AND there is at least one empty box".
This is the Venn diagram I was going by:
The pink part is what we're looking for.
