document.write( "Question 1194733: An elevator in a building starts with six passengers
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Algebra.Com's Answer #826985 by Edwin McCravy(20056) $\"\"$ $\"About$
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document.write( "We will assume that the 6 passengers are indistinguishable.  They aren't, but\r\n" );
document.write( "the probability is the same whether they are considered indistinguishable or\r\n" );
document.write( "not, and it's easier to consider them indistinguishable.\r\n" );
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document.write( "We will also assume that all 6 passengers get on the elevator at the same time\r\n" );
document.write( "in the basement (parking level, not considered a floor) and the 8 floors are\r\n" );
document.write( "numbered 1 (street level), 2, 3, 4, 5, 6, 7, and 8. \r\n" );
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document.write( "A successful elevator trip to the top (8th) floor is one in which no two\r\n" );
document.write( "passengers get off on the same floor. So we have 8 floors to pick 6 from for\r\n" );
document.write( "each single passenger to get off at. That's '8 choose 6' or C(8,6) = 28 ways.\r\n" );
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document.write( "For the number of ways any number of passengers can get off at any floor, we\r\n" );
document.write( "make a string of 6 stars to represent the 6 passengers: ******\r\n" );
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document.write( "Then we place 7 bars (1 less than the number of floors) among them, say, like\r\n" );
document.write( "this random case:\r\n" );
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document.write( " |*|**||*|*||*\r\n" );
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document.write( "This particular random arrangement of stars and bars represents the case where:\r\n" );
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document.write( "(a) The number of stars to the left of the 1st bar represents the number of\r\n" );
document.write( "passengers who get off on the 1st floor. In this case, there are no stars to the\r\n" );
document.write( "left of the first bar, so in this case, 0 passengers get off on the 1st floor.\r\n" );
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document.write( "(b) The number of stars between the 1st and 2nd bars represents the number of\r\n" );
document.write( "passengers who get off on the 2nd floor. In this case, there is only 1 star\r\n" );
document.write( "between them, so in this case, only 1 passenger gets off on the 2nd floor.\r\n" );
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document.write( "(c) The number of stars between the 2nd and 3rd bars represents the number of\r\n" );
document.write( "passengers who get off on the 3rd floor. In this case, there are 2 stars between\r\n" );
document.write( "them, so in this case, 2 passengers get off on the 3rd floor.\r\n" );
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document.write( "(d) The number of stars between the 3rd and 4th bars represents the number of\r\n" );
document.write( "passengers who get off on the 4th floor. In this case, there are no stars\r\n" );
document.write( "between them, so in this case, 0 passengers get off on the 4th floor.\r\n" );
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document.write( "(e) The number of stars between the 4th and 5th bars represents the number of\r\n" );
document.write( "passengers who get off on the 5th floor. In this case, there is only 1 star\r\n" );
document.write( " between them, so in this case, only 1 passenger gets off on the 5th floor.\r\n" );
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document.write( "(f) The number of stars between the 5th and 6th bars represents the number of\r\n" );
document.write( "passengers who get off on the 6th floor. In this case, there is only 1 star\r\n" );
document.write( "between them, so in this case, 1 passenger gets off on the 6th floor.\r\n" );
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document.write( "(g) The number of stars between the 6th and 7th bars represents the number of\r\n" );
document.write( "passengers who get off on the 7th floor. In this case, there is only 1 star\r\n" );
document.write( "between them, so in this case, only 1 passenger gets off on the 7th floor.\r\n" );
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document.write( "(h) The number of stars to the right of the 7th bar represents the number of\r\n" );
document.write( "passengers who get off on the 8th floor. In this case, there is only 1 star to\r\n" );
document.write( "the right of the 7th bar, so in this case, 1 passenger (the last one), gets off\r\n" );
document.write( "on the 8th floor.\r\n" );
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document.write( "******||||||| is the case when all 6 passengers get off on the 1st floor.\r\n" );
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document.write( "**|**|**||||| is the case where 2 get off on each of the first 3 floors and no\r\n" );
document.write( "passenger goes any higher. \r\n" );
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document.write( "|||||||****** is the case where all 6 passengers get off on the 8th floor.\r\n" );
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document.write( "|||***|***||| is the case where 3 get off on the 4th floor and the other 3 get\r\n" );
document.write( "off on the 5th floor.\r\n" );
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document.write( "For every case, there are 6+7=13 things in a row, counting bars and stars.\r\n" );
document.write( "There are 6 indistinguishable stars and 7 indistinguishable bars, so the\r\n" );
document.write( "number of cases of passengers getting off is 13!/(6!*7!) = 1716.\r\n" );
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document.write( "Answer: 28 successful elevator trips out of 1716 possibilities, or 28/1716 =\r\n" );
document.write( "7/429. \r\n" );
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document.write( "Edwin
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