document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=701448>Question 701448</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Are there more seven digit numbers with a 1 in them than without a 1 in them? Explain. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #432456 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=432456\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A more general question would be:<BR>\n" );
document.write( "Are there more n-digit numbers with a 1 in them than without a 1 in them?<BR>\n" );
document.write( "The fraction of n-digit numbers without a 1 in them<BR>\n" );
document.write( "(the probability that 1 would not be any of the n digits) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5Bn%5D=%288%2F9%29%289%2F10%29%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"b%5Bn%5D=%288%2F9%29%289%2F10%29%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "because the first digit cannot be zero,<BR>\n" );
document.write( "and it would not be 1 for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2F9 ALIGN=MIDDLE  ALT=\"8%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the possible digits,<BR>\n" );
document.write( "while the other <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n-1%29 ALIGN=MIDDLE  ALT=\"%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > digits have no restriction,<BR>\n" );
document.write( "and would not be 1 in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2F10 ALIGN=MIDDLE  ALT=\"9%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the cases.<BR>\n" );
document.write( "If you were taught the name you would recognize that as a<BR>\n" );
document.write( "\"geometric sequence\" or \"geometric progression\".<BR>\n" );
document.write( "Even if you did not know of a name for them,<BR>\n" );
document.write( "it is clear that the fraction of n-digit numbers without a 1 in them<BR>\n" );
document.write( "(or the probability of an n-digit number not having a 1)<BR>\n" );
document.write( "decreases as towards zero as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > increases.<BR>\n" );
document.write( "It is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2F9 ALIGN=MIDDLE  ALT=\"8%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > for 1-digit numbers,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%288%2F9%29%289%2F10%29=8%2F10 ALIGN=MIDDLE  ALT=\"%288%2F9%29%289%2F10%29=8%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" > for 2-digit numbers,<BR>\n" );
document.write( "and keeps getting smaller as we keep including more <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%289%2F10%29%3C1 ALIGN=MIDDLE  ALT=\"%289%2F10%29%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors.<BR>\n" );
document.write( "For 7-digit numbers the fraction of them without a 1 is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%288%2F9%29%289%2F10%29%5E6=0.472392%3C1%2F2 ALIGN=MIDDLE  ALT=\"%288%2F9%29%289%2F10%29%5E6=0.472392%3C1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28more%29 ALIGN=MIDDLE  ALT=\"highlight%28more%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > 7-digit numbers with a 1 in them than without a 1 in them.<BR>\n" );
document.write( " <BR>\n" );
document.write( "EXTRAS:<BR>\n" );
document.write( "Calculating <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B6%5D ALIGN=MIDDLE  ALT=\"b%5B6%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, I would find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B6%5D%3E1%2F2 ALIGN=MIDDLE  ALT=\"b%5B6%5D%3E1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "but how would I find the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5Bn%5D ALIGN=MIDDLE  ALT=\"b%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes smaller than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > without calculating and tabulating <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5Bn%5D ALIGN=MIDDLE  ALT=\"b%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > values for n=1, 2, 3, ...?<BR>\n" );
document.write( "That's where you use logarithms.<BR>\n" );
document.write( "Sorry, I'm getting carried away.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );