document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=16539>Question 16539</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In how many ways can 6 men and 6 Women be seated together, if no two women were to sit next to each other ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #8074 by </B><A HREF=/tutors/aboutme.mpl?userid=AnlytcPhil><B>AnlytcPhil(1806)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=AnlytcPhil><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=AnlytcPhil><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=8074\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In how many ways can 6 men and 6 Women be seated together, if no two women<BR>\n" );
document.write( "were to sit next to each other.<BR>\n" );
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document.write( "It's easy to see that no two men can sit next to each other either.  Thus men<BR>\n" );
document.write( "and women must alternate seating positions <BR>\n" );
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document.write( "We can have either of these two types of arrangements <BR>\n" );
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document.write( "MWMWMWMWMWMW<BR>\n" );
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document.write( "or<BR>\n" );
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document.write( "WMWMWMWMWMWM<BR>\n" );
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document.write( "We only need to consider one of these and then multiply the answer by 2.<BR>\n" );
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document.write( "The first one<BR>\n" );
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document.write( "MWMWMWMWMWMW<BR>\n" );
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document.write( "There are 6 choices for the leftmost M<BR>\n" );
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document.write( "For each of these 6 choices for the M to seat in the 1st seat, there are 6<BR>\n" );
document.write( "choices for the W to seat in the 2nd seat.  That's 6×6 ways to seat the<BR>\n" );
document.write( "leftmost MW.<BR>\n" );
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document.write( "For each of these 6×6 choices for the first MW in the leftmost 2 seats, there<BR>\n" );
document.write( "are 5 choices for the M to seat in the 3rd seat.  That's 6×6×5 ways to form<BR>\n" );
document.write( "the leftmost MWM.<BR>\n" );
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document.write( "For each of these 6×6×5 choices for the first MWM in the leftmost 3 seats,<BR>\n" );
document.write( "there are 5 choices for the W to seat in the 4th seat.  That's 6×6×5×5 ways to<BR>\n" );
document.write( "seat the leftmost MWMW.<BR>\n" );
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document.write( "For each of these 6×6×5×5 choices for the first MWMW in the leftmost 4 seats,<BR>\n" );
document.write( "there are 4 choices for the M to seat in the 5th seat.  That's 6×6×5×5×4 ways<BR>\n" );
document.write( "to form the leftmost MWMWM.<BR>\n" );
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document.write( "For each of these 6×6×5×5×4 choices for the first MWMWM in the leftmost 5<BR>\n" );
document.write( "seats, there are 4 choices for the W to seat in the 6th seat.  That's<BR>\n" );
document.write( "6×6×5×5×4×4 ways to form the leftmost MWMWMW.<BR>\n" );
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document.write( "For each of these 6×6×5×5×4×4 choices for the first MWMWMW in the leftmost 6<BR>\n" );
document.write( "seats, there are 3 choices for the M to seat in the 7th seat.  That's<BR>\n" );
document.write( "6×6×5×5×4×4×3 ways to form the leftmost MWMWMWM.<BR>\n" );
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document.write( "For each of these 6×6×5×5×4×4×3 choices for the first MWMWMWM in the leftmost<BR>\n" );
document.write( "7 seats, there are 3 choices for the W to seat in the 8th seat.  That's<BR>\n" );
document.write( "6×6×5×5×4×4×3×3 ways to form the leftmost MWMWMWMW.<BR>\n" );
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document.write( "For each of these 6×6×5×5×4×4×3×3 choices for the first MWMWMWMW in the<BR>\n" );
document.write( "leftmost 8 seats, there are 2 choices for the M to seat in the 9th seat.<BR>\n" );
document.write( "That's 6×6×5×5×4×4×3×3×2 ways to form the leftmost MWMWMWMWM.<BR>\n" );
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document.write( "For each of these 6×6×5×5×4×4×3×3×2 choices for the first MWMWMWMWM in the<BR>\n" );
document.write( "leftmost 9 seats, there are 2 choices for the W to seat in the 10th seat.<BR>\n" );
document.write( "That's 6×6×5×5×4×4×3×3×2×2 ways to form the leftmost MWMWMWMWMW.<BR>\n" );
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document.write( "For each of these 6×6×5×5×4×4×3×3×2×2 choices for the first MWMWMWMWMW in the<BR>\n" );
document.write( "leftmost 10 seats, there is only 1 choice for the M to seat in the 11th seat.<BR>\n" );
document.write( "That's 6×6×5×5×4×4×3×3×2×2×1 ways to form the leftmost MWMWMWMWMWM.<BR>\n" );
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document.write( "For each of these 6×6×5×5×4×4×3×3×2×2×1 choices for the first MWMWMWMWMW in<BR>\n" );
document.write( "the leftmost 11 seats, there is only 1 choice for the W to seat in the 12th<BR>\n" );
document.write( "seat.  That's 6×6×5×5×4×4×3×3×2×2×1×1 ways to form the leftmost MWMWMWMWMWM.<BR>\n" );
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document.write( "That's 518400.  But we must double this number, so the answer is 1036800.<BR>\n" );
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document.write( "Edwin<BR>\n" );
document.write( "AnlytcPhil@aol.com </SPAN>\n" );
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