document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206384>Question 1206384</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are 6 people who will sit in a row but out of them, Ronnie, will always be left of Annie and Rachel will always be right of Annie. How many row arrangements are possible?\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #843825 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=843825\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer:  <font color=red size=4>120</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Explanation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A,B,C = Ronnie, Annie, Rachel<BR>\n" );
document.write( "D,E,F = the other 3 people\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A must be to the left of B.<BR>\n" );
document.write( "C must be to the right of B.<BR>\n" );
document.write( "In other words: we must have A__B__C where 0 or more letters will go in the blanks.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Imagine that persons D,E, and F will select numbers 1 through 6 from a hat. The selections are done without replacement. The numbers refer to seating arrangement.<BR>\n" );
document.write( "1 = left most seat<BR>\n" );
document.write( "6 = right most seat<BR>\n" );
document.write( "Use the nCr combination formula (or Pascals Triangle) to determine there are p = 6C3 = 20 different ways to do this where order doesn't matter.<BR>\n" );
document.write( "Eg: Group {1,3,4} is the same as {3,1,4}\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Once persons D,E, and F find their seat, there are q = 3! = 3*2*1 = 6 ways to permute the 3 members of that group.<BR>\n" );
document.write( "Think of it like a game of musical chairs except none of the chairs get taken away and everyone is able to get a seat.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Therefore, we have p*q = 20*6 = <font color=red>120</font> ways to ensure that the ordering is A,B,C where there may or may not be a gap between Ronnie, Annie, Rachel.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A few examples of valid arrangements:<BR>\n" );
document.write( "ABC,DEF<BR>\n" );
document.write( "ABD,CEF<BR>\n" );
document.write( "ADB,ECF\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Examples that aren't valid arrangements<BR>\n" );
document.write( "CAB,DEF<BR>\n" );
document.write( "CAD,FEB<BR>\n" );
document.write( "Both aren't valid because C is not to the right of B.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );