document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=277145>Question 277145</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how many six digit numbers...\r<BR>\n" );
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document.write( "a. contain 7 at least once\r<BR>\n" );
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document.write( "b. start and end with a 7\r<BR>\n" );
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document.write( "c. have alternating 7 and non 7 digits </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #201824 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20055)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=201824\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "how many six digit numbers... \r\n" );
document.write( "a. contain 7 at least once \r\n" );
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document.write( "We will find the total number of 6-digit numbers and then we\r\n" );
document.write( "will subtract from that the total number of 6-digit numbers\r\n" );
document.write( "which do not contain 7 at all, then what we get will be the\r\n" );
document.write( "number of 6-digit numbers that contain 7 at lease once.\r\n" );
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document.write( "1. First we find the total number of 6-digit numbers.\r\n" );
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document.write( "Take this sampole 6-digit number:\r\n" );
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document.write( "825572 \r\n" );
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document.write( "We can choose a digit to go where the 8 is 9 ways (only 9 \r\n" );
document.write( "ways since we CANNOT choose 0 for the first digit).\r\n" );
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document.write( "We can then choose a digit to go where the first 2 is 10 ways (we CAN\r\n" );
document.write( "choose 0).\r\n" );
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document.write( "We can then choose a digit to go where the first 5 is 10 ways (we CAN\r\n" );
document.write( "choose 0).\r\n" );
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document.write( "We can then choose a digit to go where the second 5 is 10 ways (we CAN\r\n" );
document.write( "choose 0).\r\n" );
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document.write( "We can then choose a digit to go where the 7 is 10 ways (we CAN\r\n" );
document.write( "choose 0).\r\n" );
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document.write( "We can then choose a digit to go where the last digit 2 is 10 ways (we CAN\r\n" );
document.write( "choose 0).\r\n" );
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document.write( "So there are 9x10x10x10x10x10 or 900000 6-digit numbers.\r\n" );
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document.write( "[Note: we could have done that part this easier way:\r\n" );
document.write( "The largest 6 digit number is 999999. That means there are\r\n" );
document.write( "999999 positive integers from 1 to 999999, but the integers\r\n" );
document.write( "from 1 to 99999 have less than 6 digits, so to find the\r\n" );
document.write( "number of 6-digit numbers we subract 999999-99999=900000.\r\n" );
document.write( "But either way we get the same answer 900000.]\r\n" );
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document.write( "2. Now we calculate the number of 6 digit numbers that do not\r\n" );
document.write( "contain a 7\r\n" );
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document.write( "Take this sample 6-digit number which does not contain a 7\r\n" );
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document.write( "899168\r\n" );
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document.write( "We can choose a digit to go where the first 8 is 8 ways (only 8 \r\n" );
document.write( "ways since we CANNOT choose 0 or 7 for the first digit).\r\n" );
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document.write( "We can then choose a digit to go where the first 9 is 9 ways (we CAN\r\n" );
document.write( "choose 0 but not 7).\r\n" );
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document.write( "We can then choose a digit to go where the second 9 is 9 ways (we CAN\r\n" );
document.write( "choose 0 but not 7).\r\n" );
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document.write( "We can then choose a digit to go where the 1 is 9 ways (we CAN\r\n" );
document.write( "choose 0 but not 7).\r\n" );
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document.write( "We can then choose a digit to go where the 6 is 9 ways (we CAN\r\n" );
document.write( "choose 0 but not 7).\r\n" );
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document.write( "We can then choose a digit to go where the last digit 8 is 9 ways (we CAN\r\n" );
document.write( "choose 0 but not 7).\r\n" );
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document.write( "So there are 8x9x9x9x9x9 or 472392 6-digit numbers that do not contain 7.\r\n" );
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document.write( "3.  Now we subtract 472392 from 900000 and get 427608\r\n" );
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document.write( "------------------------\r\n" );
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document.write( "b. start and end with a 7 \r\n" );
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document.write( "Take this sample 6-digit number which starts and ends with a 7\r\n" );
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document.write( "734767\r\n" );
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document.write( "We can choose a digit to go where the first 7 is 1 way (we MUST \r\n" );
document.write( "choose it to be 7).\r\n" );
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document.write( "We can then choose a digit to go where the 3 is 10 ways (we CAN\r\n" );
document.write( "choose 0).\r\n" );
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document.write( "We can then choose a digit to go where the 4 is 10 ways (we CAN\r\n" );
document.write( "choose 0).\r\n" );
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document.write( "We can then choose a digit to go where the second 7 is 10 ways (we CAN\r\n" );
document.write( "choose 0).\r\n" );
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document.write( "We can then choose a digit to go where the 6 is 10 ways (we CAN\r\n" );
document.write( "choose 0).\r\n" );
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document.write( "We can choose a digit to go where the last 7 is 1 way (we MUST \r\n" );
document.write( "choose it to be 7).\r\n" );
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document.write( "So there are 1x10x10x10x10x1 or 10000  6-digit numbers that do not contain 7.\r\n" );
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document.write( "c. have alternating 7 and non 7 digits.\r\n" );
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document.write( "Ther can be like this:\r\n" );
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document.write( "747379 where the 7's go 1st, 3rd and 5th\r\n" );
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document.write( "The number of way is 1x9x1x9x1x9 or 729 \r\n" );
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document.write( "or like this:\r\n" );
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document.write( "579717 where the 7's go 2nd 4th and 6th.\r\n" );
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document.write( "The number of way is 8x1x9x1x9x1 or 648 \r\n" );
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document.write( "Answer = 729+648=1377\r\n" );
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document.write( "Edwin</PRE><BR>\n" );
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