document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210172>Question 1210172</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given a regular octagon, in how many ways can we color one diagonal red and another diagonal blue so that the two colored diagonals intersect at an endpoint?  Consider rotations and reflections distinct.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851445 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=851445\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem systematically.\r<BR>\n" );
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document.write( "**1. Total Diagonals in an Octagon**\r<BR>\n" );
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document.write( "* An octagon has 8 vertices.<BR>\n" );
document.write( "* The number of diagonals in an $n$-sided polygon is given by $n(n-3)/2$.<BR>\n" );
document.write( "* For an octagon, the number of diagonals is $8(8-3)/2 = 8(5)/2 = 20$.\r<BR>\n" );
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document.write( "**2. Choosing the Red Diagonal**\r<BR>\n" );
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document.write( "* We can choose any of the 20 diagonals to be red.\r<BR>\n" );
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document.write( "**3. Choosing the Blue Diagonal (Intersecting at an Endpoint)**\r<BR>\n" );
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document.write( "* Once we've chosen a red diagonal, we need to choose a blue diagonal that intersects it at an endpoint.<BR>\n" );
document.write( "* A diagonal has two endpoints.<BR>\n" );
document.write( "* For each endpoint of the red diagonal, we need to count how many other diagonals intersect at that endpoint.\r<BR>\n" );
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document.write( "* **Case 1: Red Diagonal is a Side-Skipping Diagonal (e.g., skips one vertex)**<BR>\n" );
document.write( "    * Consider a red diagonal that skips one vertex.<BR>\n" );
document.write( "    * At each endpoint, there are 3 other diagonals that intersect.<BR>\n" );
document.write( "    * In total, there are 3 + 3 = 6 diagonals that intersect the red diagonal.\r<BR>\n" );
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document.write( "* **Case 2: Red Diagonal is a Diameter-Type Diagonal (e.g., skips three vertices)**<BR>\n" );
document.write( "    * Consider a red diagonal that skips three vertices.<BR>\n" );
document.write( "    * At each endpoint, there are 3 other diagonals that intersect.<BR>\n" );
document.write( "    * In total, there are 3 + 3 = 6 diagonals that intersect the red diagonal.\r<BR>\n" );
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document.write( "**4. Counting the Pairs**\r<BR>\n" );
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document.write( "* Regardless of the type of red diagonal, there are always 6 diagonals that intersect it at an endpoint.<BR>\n" );
document.write( "* So, for each of the 20 red diagonals, we have 6 choices for the blue diagonal.<BR>\n" );
document.write( "* Thus, we have $20 \times 6 = 120$ pairs of diagonals.\r<BR>\n" );
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document.write( "**5. Accounting for Color Order**\r<BR>\n" );
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document.write( "* Since we're coloring one diagonal red and the other blue, the order matters. We need to multiply by 2 to account for the cases where the red and blue diagonals are swapped.\r<BR>\n" );
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document.write( "* Total ways = $120 \times 1 = 120$.\r<BR>\n" );
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document.write( "**Therefore, there are 120 ways to color one diagonal red and another diagonal blue so that the two colored diagonals intersect at an endpoint.**<BR>\n" );
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