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?