document.write( "Question 853359: eight students will be placed randomly in 12 desks on a row. what is the probability that no two empty desks will be adjacent ? \n" ); document.write( "

Algebra.Com's Answer #514079 by AnlytcPhil(1806) $\"\"$ $\"About$

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Question 853359
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document.write( "First we calculate the number of possible ways to\r\n" );
document.write( "choose 8 seats for students to sit in.  Then we'll divide\r\n" );
document.write( "by the number of ways the 8 students can sit in any seat. \r\n" );
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document.write( "Suppose the 12 seats are represented by these 12 lines:\r\n" );
document.write( "_ _ _ _ _ _ _ _ _ _ _ _\r\n" );
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document.write( "Case 1.  A student will sit in the first seat\r\n" );
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document.write( "There must be a student in every other seat, let\r\n" );
document.write( "S represent a student.  So we must have students\r\n" );
document.write( "in these 6 positions\r\n" );
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document.write( "S _ S _ S _ S _ S _ S _\r\n" );
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document.write( "But we have 8 students; 2 more to seat. In this case \r\n" );
document.write( "we can pick two of the 6 vacant seats to seat two \r\n" );
document.write( "other 2 students. The number of ways to \r\n" );
document.write( "choose these two seats is C(6,2) = 15 ways. \r\n" );
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document.write( "Case 2.  No student sits in the first seat.\r\n" );
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document.write( "As before, there must be a student in every other seat, So we\r\n" );
document.write( "must have students in these seats.  \r\n" );
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document.write( "_ S _ S _ S _ S _ S _ S\r\n" );
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document.write( "However in this case there are only 5 choices of seats for \r\n" );
document.write( "the two remaining students, since in this case the first\r\n" );
document.write( "seat must be vacant.  So the number of ways to choose the\r\n" );
document.write( "2 seats for two other students to sit in is C(5,2) = 10 ways. \r\n" );
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document.write( "That makes 15+10 or 25 ways to choose 8 seats for students to \r\n" );
document.write( "sit in:\r\n" );
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document.write( "Now we must divide that by the number of ways that any of the 8 students\r\n" );
document.write( "can sit anywhere which is:\r\n" );
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document.write( "We can choose the 8 seats any of C(12,8)\r\n" );
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document.write( "So the probability is 25/495 = 5/99\r\n" );
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document.write( "Edwin

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