document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210184>Question 1210184</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I can only remember a seven-digit telephone number if the first three digits (the \"prefix\") are equal to the next three digits or the last three digits. For example, I can remember 389-3892 and 274-9274.\r<BR>\n" );
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document.write( "How many seven-digit telephone numbers can I remember?\r<BR>\n" );
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document.write( "(For this problem, a telephone number cannot start with a 0.)<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 #851501 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=851501\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem into two cases based on the given condition.\r<BR>\n" );
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document.write( "**Case 1: The first three digits equal the next three digits.**\r<BR>\n" );
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document.write( "1.  **First three digits:** Since the first digit cannot be 0, there are 9 choices for the first digit (1-9). For the second and third digits, there are 10 choices each (0-9).<BR>\n" );
document.write( "    * Number of choices for the first three digits: 9 * 10 * 10 = 900<BR>\n" );
document.write( "2.  **Next three digits:** These digits must be the same as the first three digits, so there's only 1 choice for each of these three digits.<BR>\n" );
document.write( "    * Number of choices for the next three digits: 1<BR>\n" );
document.write( "3.  **Last digit:** There are 10 choices for the last digit (0-9).<BR>\n" );
document.write( "    * Number of choices for the last digit: 10<BR>\n" );
document.write( "4.  **Total numbers in Case 1:** 900 * 1 * 10 = 9000\r<BR>\n" );
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document.write( "**Case 2: The first three digits equal the last three digits.**\r<BR>\n" );
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document.write( "1.  **First three digits:** Same as Case 1, there are 900 choices.<BR>\n" );
document.write( "    * Number of choices for the first three digits: 900<BR>\n" );
document.write( "2.  **Next digit:** There are 10 choices for the next digit (0-9).<BR>\n" );
document.write( "    * Number of choices for the next digit: 10<BR>\n" );
document.write( "3.  **Last three digits:** These digits must be the same as the first three digits, so there's only 1 choice for each of these three digits.<BR>\n" );
document.write( "    * Number of choices for the last three digits: 1<BR>\n" );
document.write( "4.  **Total numbers in Case 2:** 900 * 10 * 1 = 9000\r<BR>\n" );
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document.write( "**Overlapping Case**\r<BR>\n" );
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document.write( "We've counted the numbers where the first three digits are equal to both the next three and the last three twice (e.g., 123-123123). We need to subtract these numbers once to avoid overcounting.\r<BR>\n" );
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document.write( "1.  **First three digits:** 900 choices.<BR>\n" );
document.write( "2.  **Next three digits:** 1 choice.<BR>\n" );
document.write( "3.  **Last three digits:** 1 choice.<BR>\n" );
document.write( "4.  **Total numbers in the overlap:** 900 * 1 * 1 = 900\r<BR>\n" );
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document.write( "**Final Calculation**\r<BR>\n" );
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document.write( "* Total numbers = (Numbers in Case 1) + (Numbers in Case 2) - (Numbers in the overlap)<BR>\n" );
document.write( "* Total numbers = 9000 + 9000 - 900 = 17100\r<BR>\n" );
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document.write( "**Therefore, you can remember 17,100 seven-digit telephone numbers.**<BR>\n" );
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