document.write( "Question 1040190: How many different ways can five boys and five girls form a circle with boys and girls alternate? \n" ); document.write( "

Algebra.Com's Answer #655046 by Edwin McCravy(20059) $\"\"$ $\"About$

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How many different ways can five boys and five girls form a circle
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document.write( "First we find the number of ways to place the boys in a circle.\r\n" );
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document.write( "Suppose the boys are A,B,C,D,E.  Then since a circle can be turned at\r\n" );
document.write( "any angle without changing the circle, we can think of, say A, being \r\n" );
document.write( "fixed, say facing north, and we simply arrange the 4 boys B,C,D,E \r\n" );
document.write( "around him. \r\n" );
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document.write( "So we can place the 4 boys in these 4 positions, \r\n" );
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document.write( "1. the first position clockwise from A, \r\n" );
document.write( "2. the first position counter-clockwise from A,\r\n" );
document.write( "3. the second position clockwise from A, \r\n" );
document.write( "4. the second position counter-clockwise from A,\r\n" );
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document.write( "So we can place the four boys around A in 4! ways.\r\n" );
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document.write( "[We do not include cases of putting 4 boys arranged around B, \r\n" );
document.write( "or around C, etc., because they are all counted among the 4! \r\n" );
document.write( "because they are just rotations of the same ones where the \r\n" );
document.write( "four were arranged around A.]\r\n" );
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document.write( "Now for each of the 4! ways to arrange the boys, there are 5\r\n" );
document.write( "positions to put the 5 girls in.  That's 5! ways. Therefore\r\n" );
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document.write( "Answer:  4!5! = 24(120) = 2880 ways. \r\n" );
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document.write( "Edwin

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