document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=842435>Question 842435</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Four men and eight women are to line up for a picture with the four men in the middle. How many ways can this be done? (Assume there are four women on each side of the group of men) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #507582 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=507582\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the 4 men in the middle can be arranged 4! ways.<BR>\n" );
document.write( "the 8 women all in a row can be arranged in 8! ways.<BR>\n" );
document.write( "the total number of ways will be 4! * 8! = 967,680 ways<BR>\n" );
document.write( "you take the 8 women in a row and split them in half and stick the 4 men in the middle.<BR>\n" );
document.write( "4! is equal to 24<BR>\n" );
document.write( "8! is equal to 40320.<BR>\n" );
document.write( "for each of the 40320 arrangements of the women, you can stick 24 arrangements of the men in the middle.<BR>\n" );
document.write( "this makes the total number of arrangements equal to 24 * 40320 which is equal to 967680.\r<BR>\n" );
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document.write( "let's assume smaller numbers so you can see what's happening.<BR>\n" );
document.write( "assume 2 men and 4 women.<BR>\n" );
document.write( "the men can be arranged in 2! ways which is equal to 2.<BR>\n" );
document.write( "the women can be arranged in 4! ways which is equal to 24.<BR>\n" );
document.write( "the total ways they can be arranged is 2 * 24 = 48.<BR>\n" );
document.write( "let letters represent the men and let number represent the women.<BR>\n" );
document.write( "the letters will be a and b.<BR>\n" );
document.write( "the numbers will be 1 and 2 and 3 and 4.<BR>\n" );
document.write( "the men can be arranged as:<BR>\n" );
document.write( "ab <BR>\n" );
document.write( "ba<BR>\n" );
document.write( "the women can be arranged as:\r<BR>\n" );
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document.write( "1234<BR>\n" );
document.write( "1243<BR>\n" );
document.write( "1324<BR>\n" );
document.write( "1342<BR>\n" );
document.write( "1423<BR>\n" );
document.write( "1432<BR>\n" );
document.write( "2134<BR>\n" );
document.write( "2143<BR>\n" );
document.write( "2314<BR>\n" );
document.write( "2341<BR>\n" );
document.write( "2413<BR>\n" );
document.write( "2431<BR>\n" );
document.write( "3124<BR>\n" );
document.write( "3142<BR>\n" );
document.write( "3214<BR>\n" );
document.write( "3241<BR>\n" );
document.write( "3412<BR>\n" );
document.write( "3421<BR>\n" );
document.write( "4123<BR>\n" );
document.write( "4132<BR>\n" );
document.write( "4213<BR>\n" );
document.write( "4231<BR>\n" );
document.write( "4312<BR>\n" );
document.write( "4321\r<BR>\n" );
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document.write( "all you do now is split the 4 women down the middle and stick the 2 men in between.<BR>\n" );
document.write( "each of the 24 arrangements for the women will contain 2 arrangements for the men, so you will wind up with 2 * 24 = 48 total arrangements.\r<BR>\n" );
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document.write( "I'll show you what happens with the first arrangement of the women.\r<BR>\n" );
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document.write( "the first arrangement is:\r<BR>\n" );
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document.write( "1234\r<BR>\n" );
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document.write( "the 2 arrangements for the men are ab and ba\r<BR>\n" );
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document.write( "stick the 2 men in the middle and you get:\r<BR>\n" );
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document.write( "12 ab 34<BR>\n" );
document.write( "12 ba 34\r<BR>\n" );
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document.write( "this happens with all 24 arrangements of the women, so the total number of arrangements becomes 48.\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );