document.write( "Question 1173344: Mr.Santos has set 8 question for Math test.
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\n" ); document.write( "b.If the question is the 1st question and the most difficult one is the 8th question, How many permutations are there for the 8 questions?
\n" ); document.write( "c.If the most difficult question and the easiest one are set next to each other, How many permutations are there for 8 questions?
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Algebra.Com's Answer #798559 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "Mr.  Santos has set 8 question for Math test.\r
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\n" ); document.write( "\n" ); document.write( "a.   How many permutations are there for the 8 questions?\r
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\n" ); document.write( "\n" ); document.write( "b.   If the \"highlight%28easiest+%29\" question is the 1st question and the most difficult one is the 8th question, \r
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\n" ); document.write( "\n" ); document.write( "c.   If the most difficult question and the easiest one are set next to each other,
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\n" ); document.write( "\n" ); document.write( "d.   If the most difficult question and the easiest one cannot be set next to each other,
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document.write( "(a)  For 8 objects, there are  8! = 1*2*3*4*5*6*7*8 = 40320 perrmutations.       ANSWER\r\n" );
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document.write( "(b)  Then the easiest and the heaviest questions are fixed at their 1st and 8th positions, respectively, \r\n" );
document.write( "     and only 6 questions are really permute.\r\n" );
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document.write( "     For 6 questions, there are 6! = 1*2*3*4*5*6 = 720 permutations, in all.     ANSWER\r\n" );
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document.write( "(c)  8 items have 8! = 8*7*6*5*4*3*2*1 = 40320  permutations.\r\n" );
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document.write( "     But in this problem, we consider not all the permutations.\r\n" );
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document.write( "     We consider only THOSE, where the most difficult question and easiest question are set next to each other.\r\n" );
document.write( "    in other words, we consider and count only those permutations, where these two questions are neighbor.\r\n" );
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document.write( "    It means, that we consider these two objects as two clued together;\r\n" );
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document.write( "    so we actually consider 7 objects, and this special clued object can exist in two states (ED) or (DE), depending\r\n" );
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document.write( "    which one of the two questions, easy (E) or difficult (D), goes first.\r\n" );
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document.write( "    THEREFORE, the total number of all such possible permutations is \r\n" );
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document.write( "        2*7! = 2 * (7*6*5*4*3*2*1) = 2 * 5040 = 10080.                            ANSWER\r\n" );
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document.write( "(d)  In part (a) above, I just explained that there are  8! = 40320  permutations for 8 items.\r\n" );
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document.write( "     In part (c), I also explained that there are  2*7! = 2 * 5040 = 10080 permutations, where the most difficult and easiest questions\r\n" );
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document.write( "     go next to each other (= are neighbors).\r\n" );
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document.write( "     Hence, the number of those permutations, where these questions ARE NOT neighbors is the difference\r\n" );
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document.write( "         40320 - 10080 = 30240.                                                  ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "The problem is just solved: I answered all questions.\r
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\n" ); document.write( "\n" ); document.write( "If you want to see many similar solved problems on PERMUTATIONS, see my lessons\r
\n" ); document.write( "\n" ); document.write( "    - Introduction to Permutations\r
\n" ); document.write( "\n" ); document.write( "    - PROOF of the formula on the number of Permutations\r
\n" ); document.write( "\n" ); document.write( "    - Simple and simplest problems on permutations\r
\n" ); document.write( "\n" ); document.write( "    - Special type permutations problems \r
\n" ); document.write( "\n" ); document.write( "    - Problems on Permutations with restrictions \r
\n" ); document.write( "\n" ); document.write( "    - OVERVIEW of lessons on Permutations and Combinations\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic  \"Combinatorics: Combinations and permutations\". \r
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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "Do not forget to post your  \" THANKS \"  to me for my teaching  (!)\r
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