document.write( "Question 1208353: how many ways can 3 male students and 2 female students be arranged in 6 chairs around a round table in the following cases:
\n" ); document.write( "1) The female students are adjacent.
\n" ); document.write( "2) The female students are adjacent and the male students are adjacent.
\n" ); document.write( "3) No two female students are adjacent. \n" ); document.write( "

Algebra.Com's Answer #846717 by ikleyn(52782) $\"\"$ $\"About$

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\n" ); document.write( "how many ways can 3 male students and 2 female students be arranged in 6 chairs
\n" ); document.write( "around a round table in the following cases:
\n" ); document.write( "1) The female students are adjacent.
\n" ); document.write( "2) The female students are adjacent and the male students are adjacent.
\n" ); document.write( "3) No two female students are adjacent.
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\n" ); document.write( "\n" ); document.write( "        As this problem is posed (with one vacant chair), I may assume that
\n" ); document.write( "        you are just familiar with more simple problems without vacant chair
\n" ); document.write( "        and know the basics of placement around circular table, as well as
\n" ); document.write( "        basics of circular permutations.\r
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\n" ); document.write( "\n" ); document.write( "        I will solve the problem in this order: (1), (3), (2).\r
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document.write( "(1) For part (1), we consider 2 female students as one block (= one glued object).\r\n" );
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document.write( "    I will call this object F.\r\n" );
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document.write( "    So, we have one free (unoccupied) chair, 3 male students and the block F: they are our objects\r\n" );
document.write( "    to place them in some order around the table.\r\n" );
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document.write( "    Having round table and circular permutations around it, we can assume that the empty chair is \r\n" );
document.write( "    in position \"North\", or at \"12 o'clock\".  \r\n" );
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document.write( "    Then for other 4 objects (3 male students and block F) we have 4! = 4*3*2*1 = 24 distinguishable permutations.\r\n" );
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document.write( "    In addition, we have 2 (two) independent permutations inside the group of two blocked females.\r\n" );
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document.write( "    In all, it gives 2*24 = 48 different possible distinguishable circular permutations.    ANSWER\r\n" );
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document.write( "    At this point, part (1) is complete.\r\n" );
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document.write( "(3) For part (3), we have 3 + 2 + 1 = 6 objects (3 male students, 2 female students, and one free chair).\r\n" );
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document.write( "    For 6 objects around a table, there are (6-1)! = 5! = 120 different distinguishable circular permutations.\r\n" );
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document.write( "    From this set of permutations, we should subtract 48 permutations of part (1), where two female\r\n" );
document.write( "    are adjacent.\r\n" );
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document.write( "    Doing it, we get the ANSWER for part (3): it is 120 - 48 = 72.\r\n" );
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document.write( "(2) For part (2), we have three objects: the block of 3 male; the block of 2 female, and the empty chair.\r\n" );
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document.write( "    For three objects, there are 3-1 = 2 (two) circular permutations.\r\n" );
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document.write( "    In addition to it, there are 3! = 6 independent permutations inside the block of 3 males \r\n" );
document.write( "    and 2! = 2 independent permutations inside the block of 2 females.\r\n" );
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document.write( "    In all, it gives  2*2*6 = 24 different distinguishable circular permutations.    ANSWER\r\n" );
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