document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210161>Question 1210161</A>: <I><SPAN STYLE=\"word-break: break-all;\"> King Arthur's round table has 8 evenly spaced chairs.  In how many ways can 8 knights be seated in the chairs, if Sir Lancelot and Sir Gawain insist on being seated 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 #851416 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851416\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Treat Lancelot and Gawain as a Single Unit**\r<BR>\n" );
document.write( "\n" );
document.write( "Since Sir Lancelot and Sir Gawain insist on sitting next to each other, we can treat them as a single unit. This unit can be arranged in two ways: Lancelot-Gawain or Gawain-Lancelot.\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Arrange the Unit and the Remaining Knights**\r<BR>\n" );
document.write( "\n" );
document.write( "Now, we have 7 entities to arrange around the circular table: the Lancelot-Gawain unit and the remaining 6 knights. The number of ways to arrange $n$ distinct objects in a circle is $(n-1)!$.\r<BR>\n" );
document.write( "\n" );
document.write( "Thus, we have $(7-1)! = 6!$ ways to arrange these 7 entities.\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Account for the Arrangement of Lancelot and Gawain**\r<BR>\n" );
document.write( "\n" );
document.write( "Since Lancelot and Gawain can switch places within their unit, we need to multiply the number of arrangements by 2.\r<BR>\n" );
document.write( "\n" );
document.write( "**4. Calculate the Total Number of Arrangements**\r<BR>\n" );
document.write( "\n" );
document.write( "Total number of arrangements = (arrangements of the unit and other knights) * (arrangements of Lancelot and Gawain)\r<BR>\n" );
document.write( "\n" );
document.write( "Total number of arrangements = $6! \times 2$\r<BR>\n" );
document.write( "\n" );
document.write( "$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$\r<BR>\n" );
document.write( "\n" );
document.write( "Total number of arrangements = $720 \times 2 = 1440$\r<BR>\n" );
document.write( "\n" );
document.write( "**Therefore, there are 1440 ways for the 8 knights to be seated if Sir Lancelot and Sir Gawain insist on sitting next to each other.**<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );