document.write( "Question 1154577: Suppose all 10 men at a party throw their hats in the center of the room. Each man then randomly selects a hat. Find the probability that none of the 10 men selects his own hat. \n" ); document.write( "

Algebra.Com's Answer #777031 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "In mathematical terms, the problem sounds in this way.\r\n" );
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document.write( "We consider all possible permutations of 10 items.\r\n" );
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document.write( "The number of all such permutations is 10! = 3628800.\r\n" );
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document.write( "Then we ask ourselves : how many are there such permutations of 10 items that no one item is at its native place ?\r\n" );
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document.write( "Such permutations (arrangements) are called derangement.\r\n" );
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document.write( "About derangements, see this Wikipedia article\r\n" );
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document.write( "    https://en.wikipedia.org/wiki/Derangement\r\n" );
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document.write( "In this article, you will find some theory about this subject and the formula to calculate the number of derangement of n items.\r\n" );
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document.write( "The formula is quite complicated, but there is a Table in this reference, containing calculated values for moderate values of n.\r\n" );
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document.write( "For n= 10, the number of derangement, from this Table, is equal to 1334961.\r\n" );
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document.write( "So, there are 10! = 3628800 permutations of 10 objects, in all (it is the sample space in this problem).\r\n" );
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document.write( "Of them, there are 1334961 favorable derangement permutations.\r\n" );
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document.write( "The probability under the problems's question is, therefore, \r\n" );
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document.write( "    P =  =  = 0.3679  (approximately).     ANSWER\r\n" );
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