document.write( "Question 1206743: At a dinner party there are 10 people. They all sit at a round table. In how many ways can they sit if neither Amy nor Lucy want to sit next to Josh? I have tried this a few times and get a different answer each time. Can you help please? \n" ); document.write( "

Algebra.Com's Answer #844361 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "At a dinner party there are 10 people. They all sit at a round table.
\n" ); document.write( "In how many ways can they sit if neither Amy nor Lucy want to sit next to Josh?
\n" ); document.write( "I have tried this a few times and get a different answer each time. Can you help please?
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document.write( "In such problems, the indistinguishable arrangements are those that are obtained \r\n" );
document.write( "one from the other by a circular rotation. \r\n" );
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document.write( "For 10 people around a round table, there are 9! distinguishable arrangements \r\n" );
document.write( "(= a universal set U of arrangements).\r\n" );
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document.write( "Of them, there are 8! arrangements, where A sits next to Josh on the left of him\r\n" );
document.write( "and 8! arrangements, where A sits next to Josh on the right of him.\r\n" );
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document.write( "So, in the universal set U, there are 2*8! arrangements {A}, where A sits next to Josh.\r\n" );
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document.write( "Similarly, in the universal set U, there are 2*8! arrangements {L}, where L sits next to Josh.\r\n" );
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document.write( "These arrangements, {A} and {L}, have non-empty intersection.\r\n" );
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document.write( "This intersection has 7! arrangements {AJL} and 7! other arrangements {LJA}.\r\n" );
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document.write( "From it, we conclude that the number of favorable arrangements is \r\n" );
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document.write( "    9! - 2*8! - 2*8! + 2*7!  = 1*2*3*4*5*6*7*8*9 - 4*(1*2*3*4*5*6*7*8) + 2*(1*2*3*4*5*6*7) = 211680.    ANSWER\r\n" );
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