document.write( "Question 1160504: Eight women and five men are standing in a line.
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Algebra.Com's Answer #784054 by ikleyn(52787) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "            It is strange to me to see placed into one capsule three totally different in their complexity subjects.\r
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\n" ); document.write( "\n" ); document.write( "            Part (a) is trivial and routine;\r
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\n" ); document.write( "\n" ); document.write( "            Part (b) is trivial and routine, too;\r
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\n" ); document.write( "\n" ); document.write( "            and finally part (c) is of the totally different level of complexity.\r
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\n" ); document.write( "\n" ); document.write( "            I will answer part (c), ONLY.\r
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document.write( "Imagine that the 8 women are standing in line with intervals (gaps) between them, as shown in the Figure below.\r\n" );
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document.write( "                     * W * W * W * W * W * W * W * W *\r\n" );
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document.write( "(\" * \"s  show intervals, or gaps). You see 7 stars between \"W\"s and two stars before and after the W-line; in all, 7+2 = 9 stars.\r\n" );
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document.write( "These stars are the only places for men: Each man can occupy one and only one position at the star place.\r\n" );
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document.write( "So, we can position 8 women by 8! = 8*7*6*5*4*3*2*1 = 40320 ways.\r\n" );
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document.write( "And we can place first  man in any of nine (*)-positions;\r\n" );
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document.write( "                 second man in any of remaining eight (*)-positions;\r\n" );
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document.write( "                 third  man in any of remaining seven positions;\r\n" );
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document.write( "                 fourth . . . \r\n" );
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document.write( "             and fifth . . . \r\n" );
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document.write( "Thus, in all, there are  8!*9*8*7*6*5 = 40320*9*8*7*6*5 = 609638400 differennt arrangements satisfying given restrictions.    ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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