document.write( "Question 1114142: In how many ways can 4 gentlemen and 2 ladies be seated at a round table so that the ladies are not together? In how many of these ways will three particular gentlemen be next to each other? \n" ); document.write( "

Algebra.Com's Answer #729146 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "To solve this problem, the best way (and I think, the standard way) is to consider the whole set of permutations and
\n" ); document.write( "the complementary set of permutations, and then to take the difference.\r
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document.write( "1.  The whole set of permutations in this problem is the set of all permutations (seating arrangements) of 6 persons at \r\n" );
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document.write( "    It is well known fact that the number of all such permutations (circular permutations) is  (6-1)! = 5! = 120.\r\n" );
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document.write( "2.  The complementary set of permutations is the set, where two ladies are sitting / (seating ?) together.\r\n" );
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document.write( "    By considering this pair as one object, we have then the set of all circular permutations of 5 objects, \r\n" );
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document.write( "    which consists of (5-1)! = 4! = 24 permutations.\r\n" );
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document.write( "    We then must double this number 2*24 = 48  to distinct permutations of the type  (Alice-Beatrice) and  (Beatrice-Alice) inside these pairs.\r\n" );
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document.write( "    It gives the final answer  120 - 48 = 72.\r\n" );
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document.write( "Answer.  In how many ways can 4 gentlemen and 2 ladies be seated at a round table so that the ladies are not together? - in 72 wys.\r\n" );
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\n" ); document.write( "On permutations, and specifically on circular permutations, see the 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( "    - Problems on Permutations\r
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\n" ); document.write( "\n" ); document.write( "    - Persons sitting around a circular table (*)\r
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\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( "into your archive and use when it is needed.\r
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