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Suppose we have twelve people: six men and six women. They randomly sat around a circular table. What's the probability that both male and female groups accidentally formed a single conjoined cluster each? In other words (more accurately), what's the probability that there is no man or men between any two women and also no woman or women between any two men (whoever we choose)?

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1 Answer 1

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There are $12!$ possible seating patterns, and $6!\cdot6!$ patterns of people for each pattern of gender, so $\dfrac{12!}{6!\cdot6!}=\dbinom{12}6=924$ gender patterns. There are 12 rotations of the target gender pattern. So the probability is $\dfrac{12}{924}=\dfrac1{77}$.

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  • $\begingroup$ Thanks, but... I have two observations, if you permit. 1) Firstly, we have permutations with repetitions rather than permutations of unique objects. All six men are equal and indistinguishable in this problem , and so are all women. 2) Secondly, we have a closed system (circle) rather than an open line. Won't those two factors affect the number of permutations? I mean, the number 12! seems too big for me. If there were 12 different objects in a straight line, then yes, 12!. $\endgroup$
    – Alexander
    Nov 20, 2023 at 4:19
  • $\begingroup$ I don’t think you understood the calculations. And I’m not sure what “12! seems too big for me” even means. $\endgroup$
    – Sneftel
    Nov 20, 2023 at 6:39
  • $\begingroup$ @Alexander Treating the problem as having 12 unique objects is fine, it doesn't end up changing the answer and is easier to calculate anyway. Having a circle instead of an open line is important, that's why Sneftel multiplied by 12 possible rotations of our pattern instead of the 2 rotations a straight line would have produced. $\endgroup$ Nov 20, 2023 at 14:30
  • $\begingroup$ Thanks @all, I finally understood it. So, essentially, only 77 different variations, of which just one is the appropriate case. $\endgroup$
    – Alexander
    Nov 20, 2023 at 16:39
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    $\begingroup$ Not "essentially", no. There are 77 equivalence classes under permutations of same-gender people and rotations. There are 12! different variations overall, of which 12!/77 are appropriate. $\endgroup$
    – Sneftel
    Nov 20, 2023 at 16:42

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