document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=156377>Question 156377</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many 6 digit numbers can be made with any combination and permutation of 3 and 7?  (e.g. 377777, 737777, 777333, 733733, 333337) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #115176 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=115176\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's try to find the number of combinations that 3 and 7 can make up for a two digit number.\r<BR>\n" );
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document.write( "Let's try a smaller example. So let's use the numbers 3 and 7 to make a two digit number, three digit number, and a four digit number to see what is going on\r<BR>\n" );
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document.write( "So let's list ALL of the possible two digit numbers:\r<BR>\n" );
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document.write( "33<BR>\n" );
document.write( "37<BR>\n" );
document.write( "73<BR>\n" );
document.write( "77\r<BR>\n" );
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document.write( "So for a two digit number, there are 4 possible combinations\r<BR>\n" );
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document.write( "So let's list ALL of the possible three digit numbers:\r<BR>\n" );
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document.write( "333<BR>\n" );
document.write( "337<BR>\n" );
document.write( "373<BR>\n" );
document.write( "377<BR>\n" );
document.write( "733<BR>\n" );
document.write( "737<BR>\n" );
document.write( "773<BR>\n" );
document.write( "777\r<BR>\n" );
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document.write( "So for a three digit number, there are 8 different combinations.\r<BR>\n" );
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document.write( "So let's list ALL of the possible four digit numbers:\r<BR>\n" );
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document.write( "3333<BR>\n" );
document.write( "3337<BR>\n" );
document.write( "3373<BR>\n" );
document.write( "3377<BR>\n" );
document.write( "3733<BR>\n" );
document.write( "3737<BR>\n" );
document.write( "3773<BR>\n" );
document.write( "3777<BR>\n" );
document.write( "7333<BR>\n" );
document.write( "7337<BR>\n" );
document.write( "7373<BR>\n" );
document.write( "7377<BR>\n" );
document.write( "7733<BR>\n" );
document.write( "7737<BR>\n" );
document.write( "7773<BR>\n" );
document.write( "7777\r<BR>\n" );
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document.write( "So for a four digit number, there are 16 different combinations.\r<BR>\n" );
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document.write( "Notice how the number of combinations keeps doubling (since 2 doubles to 4, 4 doubles to 8, 8 doubles to 16, etc...). \r<BR>\n" );
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document.write( "So if we write out a 5 digit number only using 3 and 7, then we will list out 32 different combinations (which is double of 16). And finally for a six digit number, there will be 64 different combinations (which is double of 32). \r<BR>\n" );
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document.write( "Note: the quicker way (ideally once you understand what's going on) is to use this formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%5Ex ALIGN=MIDDLE  ALT=\"y=2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the number of digits and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the number of unique number combinations.\r<BR>\n" );
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document.write( "So in this case, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6 ALIGN=MIDDLE  ALT=\"x=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%5E6=64 ALIGN=MIDDLE  ALT=\"y=2%5E6=64\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So once again, we get 64 different combinations.\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So there are 64 unique 6 digit numbers that can be generated from just the two values 3 and 7.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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