document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=945583>Question 945583</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A book has 500 pages numbered 1, 2, 3… and so on. How many times does the digit 1 appear in the page numbers? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #576766 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=576766\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The digit 1 sometimes appears more than once on the same page number,<BR>\n" );
document.write( "so it appears a lot.<BR>\n" );
document.write( "It appears at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100 ALIGN=MIDDLE  ALT=\"100\"   DISABLED_style= \"max-width:90%; height:auto;\" > times as the first digit in pages 100 to 199.<BR>\n" );
document.write( "It appears as the last digit <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50 ALIGN=MIDDLE  ALT=\"50\"   DISABLED_style= \"max-width:90%; height:auto;\" > times<BR>\n" );
document.write( "(on page 1, and in other page numbers preceded by 1, 2, 3, ... 49).<BR>\n" );
document.write( "It appears as the tens digit in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > 2-digit numbers (10, 11, ...19),<BR>\n" );
document.write( "and also appears as the tens digit in many 3-digit page numbers,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > times preceded by each of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > digits 1, 2, 3,and 4.<BR>\n" );
document.write( "So, the total count is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100%2B50%2B10%2B4%2A10=100%2B50%2B50=highlight%28200%29 ALIGN=MIDDLE  ALT=\"100%2B50%2B10%2B4%2A10=100%2B50%2B50=highlight%28200%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > appearances of the digit 1.<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "That does not mean that there are 200 page numbers where the number 1 appears.<BR>\n" );
document.write( "That would be a different question.<BR>\n" );
document.write( "In that case, I would rename the page numbers as the 3-digit numbers<BR>\n" );
document.write( "001, 002, 003, .... 498, 499, 000.<BR>\n" );
document.write( "(I am using 001 for 1, 002 for 2, and so on up to 099 for 99, and I use 000 for 500).<BR>\n" );
document.write( "Those are 500 3-digit ordered arrangements,<BR>\n" );
document.write( "with 5 choices for the first digit,<BR>\n" );
document.write( "and 10 choices for each of the other two digits.<BR>\n" );
document.write( "There is a total of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2A10%2A10=500 ALIGN=MIDDLE  ALT=\"5%2A10%2A10=500\"   DISABLED_style= \"max-width:90%; height:auto;\" > such arrangements.<BR>\n" );
document.write( "How many contain the digit 1 at least once?<BR>\n" );
document.write( "How many do  not contain y=the digit 1 at all?<BR>\n" );
document.write( "If we wanted to exclude the digit 1,<BR>\n" );
document.write( "we could only use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A9%2A9=324 ALIGN=MIDDLE  ALT=\"4%2A9%2A9=324\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500 ALIGN=MIDDLE  ALT=\"500\"   DISABLED_style= \"max-width:90%; height:auto;\" > arrangements.<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500-324=176 ALIGN=MIDDLE  ALT=\"500-324=176\"   DISABLED_style= \"max-width:90%; height:auto;\" > pages contain the digit 1 at least once. </SPAN>\n" );
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