document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1116072>Question 1116072</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Ten students are to sit for an examination. Four students are doing mathematics and the other 6 different subjects. In how many ways can they be seated in a row if students doing mathematics are not to sit together? \r<BR>\n" );
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document.write( "The solution for this is 6!x7x6x5x4 but I got taught a different method and it did not work. I was taught to group all the maths people together so that there are now 7 groups. I then did 10!-7! to get the number of ways maths people not sitting together. I am really struggling with this questions and I would really appreciate it if you could post the solution and explain how you got to the answer (which is 604800). Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #731050 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=731050\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> THE WAY THE SOLUTION WAS CALCULATED:<BR>\n" );
document.write( "I have to agree with the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%21%2A7%2A6%2A5%2A4=604800 ALIGN=MIDDLE  ALT=\"5%21%2A7%2A6%2A5%2A4=604800\"   DISABLED_style= \"max-width:90%; height:auto;\" > solution.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%21=720 ALIGN=MIDDLE  ALT=\"6%21=720\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the number of ways that you can arrange <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > non-math students in a row.<BR>\n" );
document.write( "Get those students in the examination room, and let them pick spots in the row of chairs.<BR>\n" );
document.write( "Ask each of them to leave empty seats in between them.<BR>\n" );
document.write( "Remove the chairs that have no one sitting next to them.<BR>\n" );
document.write( "You have now <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2B1=7 ALIGN=MIDDLE  ALT=\"6%2B1=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > empty spots<BR>\n" );
document.write( "where you can place the math students<BR>\n" );
document.write( "in a row of 6+7=13 seats.<BR>\n" );
document.write( "As each math student walks in, let them choose among the 7 empty seats.<BR>\n" );
document.write( "There are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2A6%2A5%2A4=840 ALIGN=MIDDLE  ALT=\"7%2A6%2A5%2A4=840\"   DISABLED_style= \"max-width:90%; height:auto;\" > different ways they can choose.<BR>\n" );
document.write( "The first math student has 7 choices;<BR>\n" );
document.write( "the second one has 6;<BR>\n" );
document.write( "the third one has 5,<BR>\n" );
document.write( "and the last one has 4 choices.<BR>\n" );
document.write( " <BR>\n" );
document.write( "WHY <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%21-7%21 ALIGN=MIDDLE  ALT=\"10%21-7%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > DOES NOT WORK:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%21 ALIGN=MIDDLE  ALT=\"7%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the number of ways you can get 7 groups in a row<BR>\n" );
document.write( "so that all the members of each group are together.<BR>\n" );
document.write( "As a consequence, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%21-7%21 ALIGN=MIDDLE  ALT=\"10%21-7%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the number of ways to place the 10 students to avoid having all 4 math students together, but not preventing cluster of 3 math students seating together, or pairs of math students seating together.<BR>\n" );
document.write( " <BR>\n" );
document.write( "ANOTHER WAY:<BR>\n" );
document.write( "Out of 10 seats in a row, numbered 1 through10,<BR>\n" );
document.write( "there are only 35 ways to pick 4 non-contiguous seats for the math students:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%285%2C35%2C%0D%0A%22%23+1%22%2C%22%23+2%22%2C%22%23+3%22%2C%22%23+4%22%2C%22%23+5%22%2C%22%23+6%22%2C%22%23+7%22%2C%22%23+8%22%2C%22%23+9%22%2C%22%23+10%22%2C%0D%0A%22%23+11%22%2C%22%23+12%22%2C%22%23+13%22%2C%22%23+14%22%2C%22%23+15%22%2C%22%23+16%22%2C%22%23+17%22%2C%22%23+18%22%2C%22%23+19%22%2C%22%23+20%22%2C%0D%0A%22%23+21%22%2C%22%23+22%22%2C%22%23+23%22%2C%22%23+24%22%2C%22%23+25%22%2C%22%23+26%22%2C%22%23+27%22%2C%22%23+28%22%2C%22%23+29%22%2C%22%23+30%22%2C%0D%0A%22%23+31%22%2C%22%23+32%22%2C%22%23+33%22%2C%22%23+34%22%2C%22%23+35%22%2C%0D%0A1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C2%2C2%2C2%2C2%2C2%2C2%2C2%2C2%2C2%2C2%2C3%2C3%2C3%2C3%2C4%2C%0D%0A3%2C3%2C3%2C3%2C3%2C3%2C3%2C3%2C3%2C3%2C4%2C4%2C4%2C4%2C4%2C4%2C5%2C5%2C5%2C6%2C4%2C4%2C4%2C4%2C4%2C4%2C5%2C5%2C5%2C6%2C5%2C5%2C5%2C6%2C6%2C%0D%0A5%2C5%2C5%2C5%2C6%2C6%2C6%2C7%2C7%2C8%2C6%2C6%2C6%2C7%2C7%2C8%2C7%2C7%2C8%2C8%2C6%2C6%2C6%2C7%2C7%2C8%2C7%2C7%2C8%2C8%2C7%2C7%2C8%2C8%2C8%2C%0D%0A7%2C8%2C9%2C10%2C8%2C9%2C10%2C9%2C10%2C10%2C%0D%0A8%2C9%2C10%2C9%2C10%2C10%2C9%2C10%2C10%2C10%2C%0D%0A8%2C9%2C10%2C9%2C10%2C10%2C%0D%0A9%2C10%2C10%2C10%2C%0D%0A9%2C10%2C10%2C10%2C10%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For each of those choices, you can have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%21=24 ALIGN=MIDDLE  ALT=\"4%21=24\"   DISABLED_style= \"max-width:90%; height:auto;\" > different seating orderings for the math students, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%21=720 ALIGN=MIDDLE  ALT=\"6%21=720\"   DISABLED_style= \"max-width:90%; height:auto;\" > different seating orderings for the non-math students.<BR>\n" );
document.write( "That multiplies to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=35%2A24%2A720=604800 ALIGN=MIDDLE  ALT=\"35%2A24%2A720=604800\"   DISABLED_style= \"max-width:90%; height:auto;\" > different seating arrangements. </SPAN>\n" );
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