document.write( "Question 1194841: In how many ways can 3 boys and 3 girls be seated at a round table if each girl is to be between two boys? \n" ); document.write( "

Algebra.Com's Answer #827117 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "In how many ways can 3 boys and 3 girls be seated at a round table
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document.write( "Considering circular permutation, we can assume that the chairs are numbered consequently \r\n" );
document.write( "from 1 to 6 clockwise and the chair number \"1\" is in the fixed position \"North\".\r\n" );
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document.write( "Then, if a girl is seated at the chair \"1\", then the sequence from 1 to 6 is \"G B G B G B\".\r\n" );
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document.write( "Making permutations inside the group of girls and inside the group of boys separately,\r\n" );
document.write( "we have 3!*3!= 6*6 = 36 such different sequences, where girls occupy odd positions/chairs, \r\n" );
document.write( "while the boys occupy even positions/chairs.\r\n" );
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document.write( "Next, if a boy is seated at the chair \"1\", then the sequence from 1 to 6 is \"B G B G B G\".\r\n" );
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document.write( "Making permutations inside the group of girls and inside the group of boys separately, \r\n" );
document.write( "we have 3!*3!= 6*6 = 36 such sequences, where boys occupy odd positions/chairs, \r\n" );
document.write( "while the girls occupy even positions/chairs.\r\n" );
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document.write( "Formally, these 36 + 36 = 72 arrangements are different, since we can distinct them;\r\n" );
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document.write( "but as circular permutations, 36 seating arrangements of one type are equivalent to 36 seating arrangements of the other type,\r\n" );
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document.write( "so, actually, there are only 36 different circular permutations.\r\n" );
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document.write( "Thus, the final answer depends on which arrangements you call different.\r\n" );
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document.write( "There are 36 different circular permutations and 72 different arrangements, \r\n" );
document.write( "if we consider arrangements starting from girl or boy at the chair \"1\" as different.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "I agree with the analysis by Edwin.\r
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\n" ); document.write( "\n" ); document.write( "So, use his solution - it is correct,\r
\n" ); document.write( "\n" ); document.write( "and ignore my solution as wrong.\r
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