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


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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