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Question: In how many ways can $n$ identical balls be distributed in three green, yellow and blue colored baskets?

My asnwer: For each ball exist $3$ possibilities of choosen, so $3*3*3*...*3$ $n$ times, that is $3^n$

Book's answer: Considering that the first basket has $0$ balls, the other two can have $(0, n), (1, n-1), ..., (n, 0)$ balls, that is, there are $(n+1)$ possibilities of distribute the remaining balls between the two baskets. Considering that the first basket has $1$ ball, there are $n$ possibilities to distribute the other balls in the other baskets, and so on. So there is a total of $T = (n+1) + n + ... + 1 = (n+1)(n+2)/2$

I understand the book's answer, but why is mine wrong?

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You answer is wrong because if you put ball 1 in blue and ball 2 in green, it is the same as if you put ball 1 in green and ball 2 in blue.

You've overcounted because the balls are identical.

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  • $\begingroup$ But the baskets don't is the same, I can't see the equality on put 1 ball in blue/2 in green and 1 in green/2 in blue $\endgroup$ Dec 4, 2021 at 21:42
  • $\begingroup$ @user113581321: After you do those two operations, what is the difference? How could you possibly distinguish those two operations, given that the end results are identical? $\endgroup$ Dec 4, 2021 at 21:46
  • $\begingroup$ I see sense if it were the same baskets, but in the case of having color I can't see equality, since the distribution of the balls in combination with different colors for each basket generates organizations that can even have the balls distributed equally in number, but how do they differ in the basket color automatically generates a different combination... $\endgroup$ Dec 4, 2021 at 23:26
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    $\begingroup$ There is one ball in green basket 1 and one ball in blue basket 1. You tell me: Which was the first ball? Which was the second ball? See?... you cannot tell. $\endgroup$ Dec 4, 2021 at 23:31

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