document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1084476>Question 1084476</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose you have 10 chairs that you want to put around a circular table.<BR>\n" );
document.write( "a) If all the chairs are different colours, in how many ways can they be arranged around the table?<BR>\n" );
document.write( "This I get. Its just 1x9x8x7x6x5x4x3x2x1 which would be 362880, right?\r<BR>\n" );
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document.write( "b) If all the chairs are the same color but are numbered 1 through 10, in how many ways can they be arranged around the table if the chairs that have even numbers must be kept together.<BR>\n" );
document.write( "I don't get this at all.\r<BR>\n" );
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document.write( "c) If all the chairs are the same colour but are numbered 1 through 10, in how many ways can they be arranged around the table if the chairs numbered 1 and 2 must not be placed next to each other.<BR>\n" );
document.write( "I don't get this at all as well.\r<BR>\n" );
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document.write( "d) Suppose 4 chairs are red, 2 chairs are blue, 3 chairs are yellow and 1 chair is green. The chairs are identical expect for colour. In how many ways can they be arranged around the table?<BR>\n" );
document.write( "Again, don't get this one bit.\r<BR>\n" );
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document.write( "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 #698618 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=698618\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a) (n! / n) = (n-1)! = 362880 ways<BR>\n" );
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document.write( "b) even numbers are kept together (there are 5 even numbers)<BR>\n" );
document.write( "the 5 even numbers form a group, so the circular permutation rule tells us that there are 5 (the 5 odd numbers) + 1 group objects and they can be arranged (6-1)! = 120 ways<BR>\n" );
document.write( "but there are 5! ways to arrange the even numbers so we use the multiplication rule, therefore<BR>\n" );
document.write( "there are 120 * 120 = 14400 ways<BR>\n" );
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document.write( "c) there are 8 numbers + 1 group, using circular permutation rule there are (9-1)! = 40320 ways<BR>\n" );
document.write( "but the chairs 1 and 2 are not allowed to be next to each other, so we subtract, therefore,<BR>\n" );
document.write( "there are 9! - (2 * 8!) = 362880 - 80640 = 282240 ways<BR>\n" );
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document.write( "d) what are the restrictions on how these chairs can be placed?<BR>\n" );
document.write( "student answers that there are no restrictions, therefore<BR>\n" );
document.write( "9! / (4! * 2! * 3! * 1!) = 362880 / 288 = 1260 ways<BR>\n" );
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