document.write( "Question 824371: There are 18 pictures of Christmas Trees...\r
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Algebra.Com's Answer #496443 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The other tutor gave the probability of selecting the\r\n" );
document.write( "first 8 correctly.  But that's not what was asked. \r\n" );
document.write( "You were asked for the probability of this:\r\n" );
document.write( "After putting 18 names with 18 trees, discovering that\r\n" );
document.write( "you have placed exactly 8 names correctly with the trees \r\n" );
document.write( "and placed all remaining 10 names incorrectly with trees.\r\n" );
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document.write( "You'd have to be studying combinatorial analysis to have \r\n" );
document.write( "such a problem as this.  But maybe you are.  If so, then:\r\n" );
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document.write( "We can choose the 8 trees to get correct in C(18,8) ways.  \r\n" );
document.write( "However to get the number of ways to get all the other 10\r\n" );
document.write( "incorrect, we must make use of the formula for the number \r\n" );
document.write( "of \"derangements\".  That is, the number of permutations \r\n" );
document.write( "of a1,a2,...an such that no ak is in position k. That \r\n" );
document.write( "number is !n or \"n subfactorial\", where   \r\n" );
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document.write( "!n = ⎣(e+e-1)n!]-⎣e·n!⎦  for n ≥ 2\r\n" );
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document.write( "where ⎣ ⎦ indicates the floor function. \r\n" );
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document.write( "!10 = ⎣(e+e-1)10!]-⎣e·10!⎦ = 1334961\r\n" );
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document.write( "Number of successful ways = C(18,8)×(!10) = 43758×1334961=58415223438\r\n" );
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document.write( "Number of possible ways = 18! = 6402373705728000\r\n" );
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document.write( "So the probability of placing exactly 8 correct and exactly 10\r\n" );
document.write( "incorrect is 58415223438/6402373705728000 = 0.0000091239946499,\r\n" );
document.write( "approximately. \r\n" );
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document.write( "Number of unsuccessful ways = 6402373705728000-58415223438 = \r\n" );
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document.write( "6402315290504562\r\n" );
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document.write( "Odds in favor = 58415223438:6402315290504562\r\n" );
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document.write( "Odds against = 6402315290504562:58415223438\r\n" );
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document.write( "or odds against getting exactly 8 right and 10 wrong are about \r\n" );
document.write( "109600 to 1.\r\n" );
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document.write( "Edwin
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