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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!

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  • $\begingroup$ Is the first sentence supposed to read "A bowl contains balls with eight different colors, one of which is red."? $\endgroup$ Dec 10 '19 at 9:47
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$(1)$ 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".

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  • $\begingroup$ The phrase "one of which is red" refers to the colors of the balls in the bowl from which the balls are drawn. $\endgroup$ Dec 10 '19 at 9:36
  • $\begingroup$ @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. $\endgroup$
    – Ennar
    Dec 10 '19 at 9:42
  • $\begingroup$ 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. $\endgroup$ Dec 10 '19 at 9:46
  • $\begingroup$ 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. $\endgroup$
    – Val
    Dec 10 '19 at 9:48
  • $\begingroup$ @Ennar does that mean we add up 4 probabilities (0 to 3)? $\endgroup$
    – Val
    Dec 10 '19 at 10:04

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