# number of ways of selecting n balls from m buckets with each having different number of balls [duplicate]

Imagine you have been given n different colored balls and a task to choose at most k balls from them such that none of the same colored balls is selected for any given sequence and considering same colored balls as non-identical (distinguishable).

For example

$$5$$ Balls: Red, Red, Blue, Blue, Green. You have to choose at most $$3$$ balls.

So the solution would be:

• k = $$0$$ $$\Rightarrow$$ { $$\Phi$$ } = $$1$$ //Empty Set
• k = $$1$$ $$\Rightarrow$${(red), (red), (blue), (blue), (green)} = $$5$$
• k = $$2$$ $$\Rightarrow$$ {(red, blue), (red, blue), (red, blue), (red, blue), (red, green), (red, green), (blue, green) , (blue, green) } = $$8$$
• k = $$3$$ $$\Rightarrow$$ {(red, blue, green), (red, blue, green), (red, blue, green), (red, blue, green)} = $$4$$

So the total number of possible ways are: $$18$$

Is it possible to solve this problem using Stars and bars; Assuming 3 Buckets

• Bucket 1 = [ Red, Red]
• Bucket 2 = [ Blue, Blue]
• Bucket 3 = [ Green]

With a general approach with $$n$$ and $$K$$, Or is to possible to generalize it calculating sets where we have same colored balls and subtracting it with all possible sets.