Question 1160355
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The nine numbers can be divided into three groups:<br>
A: 1, 4, 7  (1 more than a multiple of 3)
B: 2, 5, 8  (1 less than a multiple of 3)
C: 3, 6, 9  (a multiple of 3)<br>
A sequence of three numbers, one chosen from each group, will have a sum that is divisible by 3.  Since there are the same number of numbers in each group, the arrangement of the numbers around the circle must be ABCABCABC.<br>
We can count the number of different arrangements by counting the numbers of ways we can choose the number for each position, going around the table one place at a time.<br>
(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.<br>
The total number of arrangements is the product of all the numbers of choices:<br>
9*6*3*2*2*2*1*1*1 = 1296<br>
However, two arrangements which are the same except for a rotation are considered to be the same.  That essentially means we don't know where the "starting point" is for the arrangement; and that means our count of 1296 is too large by a factor of 9.<br>
So with the given rules, the number of arrangements is 1296/9 = 144.<br>
ANSWER: 144 different arrangements<br>