We have N bottles, m red and n blue balls, n > N, m > N.
We need to know in how many ways is it possible to partition
all read and blue balls into bottles. There should be at list
one red and one blue ball in each bottle.
There is an important concept known a integer partitions
Because "There should be at least one red and one blue ball in each bottle".
We are partitioning the numbers $\displaystyle n-N~\&~m-N$
From that webpage, you can see that the is no simple formula or way of doing this problem.
Here is an example. Suppose $\displaystyle n=15,~m=10,~\&~N=6$.
We think this way. Go ahead and put a red in each bottle leaving 9 reds.
Go ahead and put a blue in each bottle leaving 4 blues.
Now there are $\displaystyle P(9,6)=26$ ways to partition 9 into 6 or fewer summands.
And $\displaystyle P(4,4)=5$ ways to partition 4 into 4 or fewer summands.
The product $\displaystyle 26\cdot 5=130$ is the number of ways of doing both.