document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210205>Question 1210205</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many 7-digit sequences have a digit that appears at least 6 times?\r<BR>\n" );
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document.write( "(For example, 3339333 and 0200000 are two such sequences. A sequence is allowed to begin with 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 #851604 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=851604\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let a 7-digit sequence be represented by $d_1 d_2 d_3 d_4 d_5 d_6 d_7$, where each $d_i \in \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$.<BR>\n" );
document.write( "We are looking for the number of sequences where a digit appears at least 6 times. This can happen in two cases:<BR>\n" );
document.write( "Case 1: A digit appears exactly 6 times.<BR>\n" );
document.write( "Case 2: A digit appears exactly 7 times.\r<BR>\n" );
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document.write( "Case 1: A digit appears exactly 6 times.<BR>\n" );
document.write( "First, choose which digit appears 6 times. There are 10 choices (0 to 9).<BR>\n" );
document.write( "Next, choose the positions for these 6 digits in the 7-digit sequence. There are $\binom{7}{6}$ ways to choose the 6 positions.<BR>\n" );
document.write( "Finally, choose the digit for the remaining position. Since this digit cannot be the same as the digit that appears 6 times, there are 9 choices for this remaining digit.<BR>\n" );
document.write( "So, the number of sequences where a digit appears exactly 6 times is $10 \times \binom{7}{6} \times 9 = 10 \times 7 \times 9 = 630$.\r<BR>\n" );
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document.write( "Case 2: A digit appears exactly 7 times.<BR>\n" );
document.write( "First, choose which digit appears 7 times. There are 10 choices (0 to 9).<BR>\n" );
document.write( "Next, all 7 positions in the sequence must be this chosen digit. There is only $\binom{7}{7} = 1$ way to place these digits.<BR>\n" );
document.write( "So, the number of sequences where a digit appears exactly 7 times is $10 \times \binom{7}{7} = 10 \times 1 = 10$.\r<BR>\n" );
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document.write( "The total number of 7-digit sequences where a digit appears at least 6 times is the sum of the numbers from Case 1 and Case 2.<BR>\n" );
document.write( "Total number of sequences = (Number of sequences with exactly 6 repetitions) + (Number of sequences with exactly 7 repetitions)<BR>\n" );
document.write( "Total number of sequences = 630 + 10 = 640.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{640}$ </SPAN>\n" );
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