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
From that webpage, you can see that the is no simple formula or way of doing this problem.
Here is an example. Suppose .
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 ways to partition 9 into 6 or fewer summands.
And ways to partition 4 into 4 or fewer summands.
The product is the number of ways of doing both.