document.write( "Question 435922: Five paychecks and envelopes are addressed to five different people. The paychecks are randomly inserted into the envelopes. What is the probability that at least one paycheck is inserted in the correct envelope? \n" ); document.write( "

Algebra.Com's Answer #301680 by Edwin McCravy(20055) $\"\"$ $\"About$

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Five paychecks and envelopes are addressed to five different people. The paychecks are randomly inserted into the envelopes. What is the probability that at least one paycheck is inserted in the correct envelope?
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document.write( "The answer is 1 minus the probability that all checks go wrong.\r\n" );
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document.write( "The number of ways n things in n original positions can be \r\n" );
document.write( "\"deranged\", that is, arranged so that none of the n things \r\n" );
document.write( "are in their original positions is a number called \r\n" );
document.write( "\"n-subfactorial\". This is written !n, and the formula for \r\n" );
document.write( "it is\r\n" );
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document.write( "Therefore the number of ways all 5 checks go wrong is\r\n" );
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document.write( "So the numerator for the probability that all the \r\n" );
document.write( "checks go wrong is 44.\r\n" );
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document.write( "The denominator is the number of different ways the \r\n" );
document.write( "checks can be placed in any envelopes is 5! or 120.\r\n" );
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document.write( "Therefore the probability that all checks go wrong \r\n" );
document.write( "is 44/120 = 11/30.\r\n" );
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document.write( "Therefore the probability that at least one went \r\n" );
document.write( "right is\r\n" );
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document.write( "1 - 11/30 = 30/30 - 11/30 = 19/30.\r\n" );
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document.write( "Edwin

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