document.write( "Question 934456: 5 boys and 3 girls are to be seated on chairs arranged in a row. If the arrangement is made by a random selection, find the probability that no two girls are selected side by side. \n" ); document.write( "
Algebra.Com's Answer #567443 by Edwin McCravy(20054)\"\" \"About 
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document.write( "We only need to consider arrangements by gender only.\r\n" );
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document.write( "We take the 3 girls and 2 boys to set up the minimum requirements\r\n" );
document.write( "for a success: \r\n" );
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document.write( "__GB__GB__G__\r\n" );
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document.write( "Now we have to place the remaining 3 boys in the 4 groups represented\r\n" );
document.write( "by the 4 blanks.  This is the number of ordered partitions of 3 things \r\n" );
document.write( "into 4 groups allowing groups of size 0.\r\n" );
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document.write( "The number of ordered partitions of n things into r groups is C(n+r-1,r-1), \r\n" );
document.write( "counting groups of 0.\r\n" );
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document.write( "So we have n=3, r=4\r\n" );
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document.write( "C(3+4-1,4-1) = C(6,3) = 20 ways to place the remaining boys into those\r\n" );
document.write( "4 blanks (including leaving some of the blanks empty).\r\n" );
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document.write( "Just for illustrative purposes, those 20 arrangements by gender only are:\r\n" );
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document.write( "1.  GBGBGBBB\r\n" );
document.write( "2.  GBGBBGBB\r\n" );
document.write( "3.  GBGBBBGB\r\n" );
document.write( "4.  GBGBBBBG\r\n" );
document.write( "5.  GBBGBGBB\r\n" );
document.write( "6.  GBBGBBGB\r\n" );
document.write( "7.  GBBGBBBG\r\n" );
document.write( "8.  GBBBGBGB\r\n" );
document.write( "9.  GBBBGBBG\r\n" );
document.write( "10.  GBBBBGBG\r\n" );
document.write( "11.  BGBGBGBB\r\n" );
document.write( "12.  BGBGBBGB\r\n" );
document.write( "13.  BGBGBBBG\r\n" );
document.write( "14.  BGBBGBGB\r\n" );
document.write( "15.  BGBBGBBG\r\n" );
document.write( "16.  BGBBBGBG\r\n" );
document.write( "17.  BBGBGBGB\r\n" );
document.write( "18.  BBGBGBBG\r\n" );
document.write( "19.  BBGBBGBG\r\n" );
document.write( "20.  BBBGBGBG\r\n" );
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document.write( "So the numerator of the desired probability is 20.\r\n" );
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document.write( "The denominator is the total number of gender arrangements.\r\n" );
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document.write( "We can pick any 3 of the 8 positions to place the 3 girls \r\n" );
document.write( "C(8,3) = 56 ways \r\n" );
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document.write( "Answer C(6,3)/C(8,3) = 20/56 = 5/14\r\n" );
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document.write( "Edwin
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