document.write( "Question 1186630: what fraction of the integers $\"highlight%28cross%28berween%29%29\"$ between 0 and 1000 include exactly two 6s? \n" ); document.write( "

Algebra.Com's Answer #817559 by ikleyn(52915) $\"\"$ $\"About$

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\n" ); document.write( "what fraction of the integers between 0 and 1000 include exactly two 6s?
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\n" ); document.write( "\n" ); document.write( " This problem has a brilliant, elegant and unexpectedly simple solution.\r
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document.write( "Consider all integer numbers from 0 to 999, inclusive.\r\n" );
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document.write( "In all, there are exactly 1000 such numbers.\r\n" );
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document.write( "Will consider one-digit numbers, like 3, 7 as three digit numbers with leading zeroes  003, 007.\r\n" );
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document.write( "Will consider two digit numbers, like 37 as three digit numbers with leading zero 037.\r\n" );
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document.write( "It will change NOTHING in the solution.\r\n" );
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document.write( "All such three-digit numbers, that have only two digits \"6\", are in the following three disjoint categories:\r\n" );
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document.write( "    - having \"6\" in the first  and the second positions, only;\r\n" );
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document.write( "    - having \"6\" in the first  and the third  positions, only;\r\n" );
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document.write( "    - having \"6\" in the second and the third positions, only.\r\n" );
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document.write( "Now, let's consider all three-digit numbers with the digits 6 in \"hundreds\" and \"tens\" position.\r\n" );
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document.write( "    The amount of such numbers is 9, obviously, because there are 9 such thre-digit numbers that have 9 possible digits \r\n" );
document.write( "    (all excepting 6)  in the \"ones\" position.\r\n" );
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document.write( "Next, consider all three-digit numbers with the digits 6 in \"hundreds\" and \"ones\" position.\r\n" );
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document.write( "    The amount of such numbers is 9, obviously, because there are 9 such thre-digit numbers that have 9 possible digits \r\n" );
document.write( "    (all excepting 6) in the \"tens\" position.\r\n" );
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document.write( "Finally, consider all three-digit numbers with the digits 6 in \"tens\" and \"ones\" position.\r\n" );
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document.write( "    The amount of such numbers is 9, obviously, because there are 9 such thre-digit numbers that have 9 possible digits \r\n" );
document.write( "    (all excepting 6) in the \"ones\" position.\r\n" );
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document.write( "ANSWER.  In all, there are 9+9+9 = 27 numbers from 0 to 1000, having only two digits \"6\" in their records.\r\n" );
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