9 balls, 3 colors

Probability Level 3

There are 9 balls in a bag - 3 green, 3 blue, and 3 red.

Three kids reach in and grab 3 random balls each.

How many different arrangements can they end up with?

Clarification: The balls are indistinguishable, and if one kid got RGR, its the same distribution as if he got GRR.

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The answer is 55.

1 solution

Geoff Pilling
May 14, 2016

If $N(a,b,c)$ is the number of ways the first kid can have $a$ matching balls, the second kid can have $b$ matching balls, and the third kid can have $c$ matching balls, then:

$N(1,1,1) = 1$

$N(1,2,2) = 6$

$N(2,1,2) = 6$

$N(2,2,1) = 6$

$N(2,2,2) = 12$

$N(3,2,2) = 6$

$N(2,3,2) = 6$

$N(2,2,3) = 6$

$N(3,3,3) = 6$

So, the total is $6*7+1+12=\boxed{55}$

I'm having some trouble understanding your notation, in particular the part about "matching balls". If a kid has one red ball, one blue ball, and one green ball, I think that counts as "1 matching ball". If a kid has one red ball and two blue balls, I think that counts as "2 matching balls".

If that's true, then $N(1,1,2) = 0$ , because this means the first two kids have one ball of each color, so the third kid must also have one ball of each color. But $N(1,2,2) = 6$ .

Jon Haussmann - 5 years ago

OOOps, your right... I had a typo in my solution... I've fixed it... Does it make more sense now?

Geoff Pilling - 5 years ago

Yes, that makes sense. One more thing: $N(2,2,2) = 12$ . That brings the total up to 55.

Jon Haussmann - 5 years ago

@Jon Haussmann – Ah shux... You are right again, Jon... @Calvin Lin any chance you could update this answer to 55? As Jon pointed out, I made a mistake in my calculation. Or maybe @Andrew Ellinor ?

Geoff Pilling - 5 years ago

@Geoff Pilling – I have updated the answer to 55.

Calvin Lin Staff - 5 years ago

9 balls, 3 colors

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The answer is 55.

1 solution

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