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Nine people sit around a round table. The number of ways of selecting four of them such that they are not from adjacent seats, is...?

MY ATTEMPT:- I placed first non selecting persons around the table. Five places are present in between these five non selecting persons. If I place four other selecting persons in any of the four places out of these five places, they will not be adjacent. So according to me answer will be $^5C_4 = 5$. But it is not the correct answer. Correct answer is $9$.

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You have three non-selected people with selected people on each side and one pair of non-selected people sitting together. Once you choose the left one of the non-selected people everything is determined and that can be any of the nine.

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  • $\begingroup$ I've got your point that one pair of non selected persons can be selected in 9 ways. But what is wrong in my method $\endgroup$ – Ruchit Vithani Jul 13 '17 at 4:50
  • $\begingroup$ The problem is that if you place the non-selected people as n_n_n_n_n_, with the _ being possible places where you could put the selected people, however you choose where to put the selected people, you will always have a non-selected person in the first place. So you are missing the ways where there is a selected person in the first place. To cover those you would start with sn_n_n_n_n, and then there are another 4 ways to complete that. $\endgroup$ – Especially Lime Jul 13 '17 at 9:02

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