# How many ways to distribute balls so that at least $4$ red balls are selected? at most $3$ red balls are selected?

I'm trying to figure out the following: A bowl contains balls with eight different colors, one of which is red. Assuming there are at least $$20$$ balls of each color, how many ways can a total of $$20$$ balls be distributed among the eight different colors if

(1) the bowl must contain at least four red balls?

(2) must contain at most $$3$$ red balls?

For (1), I got $$245,157$$. Is this correct? $$(16 + 8 - 1)C16$$

Also, any clues or hints to solve the second part would be appreciated!

• Is the first sentence supposed to read "A bowl contains balls with eight different colors, one of which is red."? Dec 10 '19 at 9:47

$$(1)$$ looks correct.

Hint: "at most 3 red balls" = "all distributions" - "at least 4 red balls".

Possibly, attention needs to be paid when counting "all distributions", since the wording "one of which is red" might be interpreted as "at least 1 red".

• The phrase "one of which is red" refers to the colors of the balls in the bowl from which the balls are drawn. Dec 10 '19 at 9:36
• @N.F.Taussig, that is how I would interpret it, but the whole sentence is awkward: "A bowl contains eight balls with different colors, one of which is red." so "one of which" could also refer to balls, not colors. If we wrote it properly as: "A bowl contains balls with eight different colors, one of which is red." it's much clearer that "one of which" refers to colors. That's why I said "possibly", since we don't have the original wording of the exercise. Dec 10 '19 at 9:42
• Upon rereading the question, I suspect it meant to say a bowl contains balls with eight different colors, one of which is red. Otherwise, the following sentence makes no sense. Dec 10 '19 at 9:46
• Thank you both for responding! It does mean "A bowl contains balls with eight different colors, one of which is red." I just got clarification.
– Val
Dec 10 '19 at 9:48
• @Ennar does that mean we add up 4 probabilities (0 to 3)?
– Val
Dec 10 '19 at 10:04