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The problem involves arranging balls of different color in a line such that no two balls of the same color are adjacent to each other. Eventually I would like to find a generalization of having $m$ different colors of balls and $n$ balls of each color, where each ball besides color are indistinguishable. But for now I will start with $3$ colors, red blue and yellow, and having $3$ of each color.

If we had an infinite number of each ball, then i know the answer would be $3(2^8)$ but this is including arrangements where we might only have red and blue balls alternating and not have any yellows.

One thought was to consider "grouping" the colors in its $3!$ permutations and finding the number of arrangements of the groups. But this wouldn't consider arrangements such as $RYRYBYBRB$

I feel like recurrence may be needed for a generalization but without being able to find this simple case I don't know if I can use it. Searching other questions of this sort doesn't help me either as it isn't this type of restriction.

Brute forcing it is one option but it seems like a mess even for this smaller case, so it would only be good for smaller cases.

Any help would be greatly appreciated

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marked as duplicate by leonbloy, Mike Earnest, Theoretical Economist, Lord Shark the Unknown, Brahadeesh Aug 21 '18 at 6:49

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  • $\begingroup$ Try to state the problem in the question body (not just in the title). The balls are placed in a row or in a circle ? $\endgroup$ – leonbloy Aug 20 '18 at 22:00
  • $\begingroup$ Oh my I forgot to add that part. These are arranged in a straight line. Will edit $\endgroup$ – WaveX Aug 20 '18 at 22:01
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    $\begingroup$ This problem covers the cases of $3$ red, $3$ blue, and $3$ green balls arranged in a row so that two consecutive balls are of the same color. $\endgroup$ – N. F. Taussig Aug 20 '18 at 22:03
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    $\begingroup$ @N.F.Taussig ... "so that no two consecutive balls..." , no? $\endgroup$ – leonbloy Aug 20 '18 at 22:05
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    $\begingroup$ See this question for possible solutions. I'm certain there are other answers elsewhere on MSE too. $\endgroup$ – N. Shales Aug 20 '18 at 22:25

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