document.write( "Question 605243: 6 boys and 5 girls are to be seated around a table. Find the number of ways that this can be done in each of the following cases.
\n" ); document.write( "a. there are no restrictions
\n" ); document.write( "b. no 2 girls are adjacent
\n" ); document.write( "c. all girls form a single block
\n" ); document.write( "d. a particular girl G is adjacent to two particular B1 and B2 \n" ); document.write( "
Algebra.Com's Answer #381599 by Edwin McCravy(20056)\"\" \"About 
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6 boys and 5 girls are to be seated around a table. Find the number of ways that this can be done in each of the following cases.\r
\n" ); document.write( "\n" ); document.write( "a. there are no restrictions
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document.write( "First we find the number of ways they could line up  in a single file to march\r\n" );
document.write( "to the table and then divide by the number of ways the table could be rotated.\r\n" );
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document.write( "They could line up in a single file in 11! ways and the table could be rotated\r\n" );
document.write( "11 ways.  So the answer is \r\n" );
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document.write( "\"11%21%2F11\" = 10! = 3628800\r\n" );
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\n" ); document.write( "b. no 2 girls are adjacent
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document.write( "Let's seat the 5 girls first.  The girls can be lined up in 5! ways and the\r\n" );
document.write( "table can be rotated 5 ways, so we can seat the girls in 5!/5 = 4! = 24 ways.\r\n" );
document.write( "There are 5 spaces between the girls.  We pick 5 of the boys to go in those 5\r\n" );
document.write( "spaces in 6C5 = 6 ways, and we can arrange those 5 boys in 5! = 120 ways.  We\r\n" );
document.write( "have one boy left over to seat, so he will have to sit next to a boy.  He can\r\n" );
document.write( "pick the boy to sit next to in 5 ways, and he can pick whether to sit on his\r\n" );
document.write( "left or his right in 2 ways, so that's\r\n" );
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document.write( "24·6·120·5·2 = 172800.\r\n" );
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\n" ); document.write( "c. all girls form a single block
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document.write( "So the boys must also form a single block.\r\n" );
document.write( "Then the single block of girls can be arranged in 5! ways.  The single block of\r\n" );
document.write( "boys can be arranged in 6!.\r\n" );
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document.write( "Answer 5!6! = 120·720 = 86400 \r\n" );
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\n" ); document.write( "d. a particular girl G is adjacent to two particular B1 and B2
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document.write( "That trio can be seated 2 ways, B1,G,B2 or B2,G, B1.  The remaining 8 people\r\n" );
document.write( "can sit anywhere.\r\n" );
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document.write( "So with B1,G,B2, we have to seat 9 \"things\" around the table, 8 singles and 1\r\n" );
document.write( "trio.  We could arrange them in a line in 9! ways, then the table could be\r\n" );
document.write( "rotated in 9 ways, so that 9!/9 = 8! = 40320 ways. That's for the trio sitting\r\n" );
document.write( "B1,G,B2. But they could also sit B2,G,B1, so that's another 40320.\r\n" );
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document.write( "Answer 8!·2 = 40320·2 = 80640 ways. \r\n" );
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document.write( "Edwin
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