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# Help plz

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Determine the number of ways of placing the numbers  in a circle, so that the sum of any three numbers in consecutive positions is divisible by  (Two arrangements are considered the same if one arrangement can be rotated to obtain the other.)
I tried and got 1296 or 432, I don't know which is right???

Jun 2, 2020
edited by Sarvajit  Jun 2, 2020

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1) We can choose any of the 9 numbers first. 9 choices.
(2) The second number must be one of the 6 numbers in the two other groups. 6 choices.
(3) The third number must be one of the 3 numbers in the third group. 3 choices.
(4) The fourth number must be one of the remaining 2 numbers in the first group. 2 choices.
(5) The fifth number must be one of the remaining 2 numbers in the second group. 2 choices.
(6) The sixth number must be one of the remaining 2 numbers in the third group. 2 choices.
(7) The seventh, eighth, and ninth numbers must be the 1 remaining numbers in the first, second, and third groups, respectively. 1 choice each.

The total number of arrangements is the product of all the numbers of choices:

9*6*3*2*2*2*1*1*1 = 1296/9=144

Jun 2, 2020
edited by Sarvajit  Jun 2, 2020
edited by Sarvajit  Jun 2, 2020