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I'm having a weird time with this one. given $n$ pairs of brothers, I expect that if i sum all the possible ways to arrange $2n$ individuals in a row, then subtract the ways to sit $n$ brothers together I would get the amount of ways to sit them separate from each other. However I get: $$2n! - n!$$

Which seems to be the wrong answer. I found that for sitting 2 pairs, we get a total of $8$ combinations, so: $$4! - 2! = 24 - 2 = 22 \not = 8$$

I think the correct answer is sitting $n$ individuals who are not brothers with spaces, than sit the $n$ remaining in those spaces. We can do this in $n$ ways so all in all we get $$n!n! \cdot n$$ Why isn't the first solution working? Or is my second solution isn't really a solution also?

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  • $\begingroup$ I don't understand the $n!$ term. You need to exclude $\textit {any}$ two brothers from sitting next to eachother. How is that given by $n!$ ? Work it out explicily for, say, $n=3$ , compare your answer(s) to that. $\endgroup$
    – lulu
    May 11, 2020 at 14:05
  • $\begingroup$ here is a duplicate. $\endgroup$
    – lulu
    May 11, 2020 at 14:08

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