document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1160355>Question 1160355</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine the number of ways of placing the numbers 1-9 in a circle, so that the sum of any three numbers in consecutive positions is divisible by 3 (Two arrangements are considered the same if one arrangement can be rotated to obtain the other.)<BR>\n" );
document.write( "please asap thank you </SPAN>\n" );
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document.write( "The nine numbers can be divided into three groups:<br><BR>\n" );
document.write( "A: 1, 4, 7  (1 more than a multiple of 3)<BR>\n" );
document.write( "B: 2, 5, 8  (1 less than a multiple of 3)<BR>\n" );
document.write( "C: 3, 6, 9  (a multiple of 3)<br><BR>\n" );
document.write( "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><BR>\n" );
document.write( "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><BR>\n" );
document.write( "(1) We can choose any of the 9 numbers first.  9 choices.<BR>\n" );
document.write( "(2) The second number must be one of the 6 numbers in the two other groups.  6 choices.<BR>\n" );
document.write( "(3) The third number must be one of the 3 numbers in the third group.  3 choices.<BR>\n" );
document.write( "(4) The fourth number must be one of the remaining 2 numbers in the first group.  2 choices.<BR>\n" );
document.write( "(5) The fifth number must be one of the remaining 2 numbers in the second group.  2 choices.<BR>\n" );
document.write( "(6) The sixth number must be one of the remaining 2 numbers in the third group.  2 choices.<BR>\n" );
document.write( "(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><BR>\n" );
document.write( "The total number of arrangements is the product of all the numbers of choices:<br><BR>\n" );
document.write( "9*6*3*2*2*2*1*1*1 = 1296<br><BR>\n" );
document.write( "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><BR>\n" );
document.write( "So with the given rules, the number of arrangements is 1296/9 = 144.<br><BR>\n" );
document.write( "ANSWER: 144 different arrangements<br><BR>\n" );
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