document.write( "Question 937392: 7 girls and 5 boys are to be seated around a table. Find the number of ways if 3 particular girls must be together, and 2 particular boys must not be adjacent? \n" ); document.write( "
Algebra.Com's Answer #571740 by Edwin McCravy(20055)\"\" \"About 
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7 girls and 5 boys are to be seated around a table.
\n" ); document.write( "Find the number of ways if 3 particular girls must be together,
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document.write( "Whenever we have a problem involving people sitting around a table,\r\n" );
document.write( "we consider it as though the table were placed on a large turntable\r\n" );
document.write( "and could be rotated clockwise or counter-clockwise with no change\r\n" );
document.write( "in the seating arrangement.\r\n" );
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document.write( "1. First we'll enumerate the ways with the 3 girls sitting together\r\n" );
document.write( "but with no restriction on the boys.\r\n" );
document.write( "2. Then we'll enumerate the cases with the 3 girls together and the 2 boys\r\n" );
document.write( "together.\r\n" );
document.write( "3. Then we'll subtract the result of 2 from the result of 1.\r\n" );
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document.write( "1. We enumerate the ways with the 3 girls sitting together\r\n" );
document.write( "but with no restriction on the boys.\r\n" );
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document.write( "The 3 girls can be arranged together in 3! = 6 ways, \r\n" );
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document.write( "So for each of those 6 ways that we can put the girls together, we can\r\n" );
document.write( "seat the remaining 9 people around the 3 girls in 9! ways.\r\n" );
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document.write( "That's (3!)(9!) = (6)(362880) = 2177280 ways.\r\n" );
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document.write( "2. We enumerate the cases with the 3 girls together and the 2 boys\r\n" );
document.write( "together.\r\n" );
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document.write( "As before, the 3 girls can be arranged together in 3! = 6 ways, \r\n" );
document.write( "The 2 boys can be arranged together in 2! = 2 ways.\r\n" );
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document.write( "So that's (3!)(2!) = (6)(2) = 12 ways to place the 3 girls together and \r\n" );
document.write( "the 2 boys together.\r\n" );
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document.write( "Now, instead of having 9 people to place around the 3 girls, we only have 8\r\n" );
document.write( "\"THINGS\", 4 individual girls, 3 individual boys, and 1 pair of boys.\r\n" );
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document.write( "So for each of those 12 ways that we can put the 3 girls and the 2 boys\r\n" );
document.write( "together, we can seat the remaining 8 THINGS around the girls in 8! ways.\r\n" );
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document.write( "That's (3!)(2!)(8!) = (6)(2)(40320) = 483840 ways to put the 3 girls together\r\n" );
document.write( "and the 2 boys together.\r\n" );
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document.write( "3. Now we subtract the result of 2 from the result of 1.\r\n" );
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document.write( "2177280 - 483840 = 1693440 ways.\r\n" );
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document.write( "Edwin
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