I am trying to solve this problem of counting the number of arrangements of 8 people around a square table, as shown in the figure below, To solve this problem you can consider arrangements obtained from rotation to be similar -:

enter image description here

The first part of the question asks how many possible arrangements of 8 people are there on this square table, my reasoning for coming up with an answer is as follows, each of the circular arrangement of 8 people around the square table corresponds to 4 linear arrangements so by this reasoning the answer I came up with $$ \frac{8!}{4} = 10080 $$ square arrangements.

The second part of the question asks me in how many square arrangements do A and B don't sit together, here is how I approached the problem, I first counted the number of linear arrangements in which A and B sit together $ 7! \cdot 2! $ and using this I counted the number of square arrangements in which A and B sit next to each other as $ \dfrac{7! \cdot 2!}{4 \cdot 2!} = 1260 $ square arrangements in which A and B sit next to each other and then I subtract this number from the total number of square arrangements $ 10080 - 1260 = 8820 $ arrangements in which A and B don't sit next to each other, I am not sure if my answer is correct but I think it should be, it would be great if someone could confirm this.

  • $\begingroup$ If I understand the first question right, the first answer is correct, but for an arbitrarily arrangement, chances for B to be beside A is 2/7, the second answer should be 10080 / 7 * 5 $\endgroup$ – Coolwater Jul 7 '14 at 11:15
  • $\begingroup$ How, can you explain it a little bit? $\endgroup$ – AnkitSablok Jul 7 '14 at 11:18
  • $\begingroup$ Using your approach there are 8 places for A, 2 for B, 6 for C, 5 for D, etc. so A is beside B in 8*2*6*5*4*3*2*1 / 4 = 2880 cases $\endgroup$ – Coolwater Jul 7 '14 at 11:21
  • $\begingroup$ how are there 2 places for B?, okk I got it the number of arrangements in which A and B are together, yeah B can be either to the right of A or to left of A, but then how do you come up with 7200 as the answer? $\endgroup$ – AnkitSablok Jul 7 '14 at 11:23
  • $\begingroup$ You know from first part there is 10080. Subtract the 2880 and you get the right answer.There are 7 other people than A in a random arrangement, 2 of them are beside A, hence 100% - 2/7 = 5/7 of the 10080 cases is also right answer. $\endgroup$ – Coolwater Jul 7 '14 at 11:34

The exclusion of arrangements that can be obtained from rotation comes to the same as the extra condition that $A$ is seated e.g. at the upper side. This because in any case there is exactly one rotation that brings him there. Then there are $2$ possibilities for $A$. The first part then gives $2\times7!=10080$ possibilities, confirming your own answer. The second part gives $2\times6\times6!=4320$ possibilities (if I understand well that $A$ and $B$ are not sitting next to eachother here). The factor $6$ corresponds with the possibilities for $B$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.