document.write( "Question 1078654: Find the sum of all positive six-digit integers whose digits are a permutation of the digits in the number 123456 without listing them all. \n" ); document.write( "

Algebra.Com's Answer #693075 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Answer. 111111*21*120.\r
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document.write( "In all, there are 6! = 1*2*3*4*5*6 = 720 such 6-digit numbers.\r\n" );
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document.write( "If you will write all these 720 6-digit numbers in one VERY long column, then\r\n" );
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document.write( "    1.  you will have 120 numbers that have \"1\" in the \"ones\" position;\r\n" );
document.write( "              120 numbers that have \"2\" in the \"ones\" position;\r\n" );
document.write( "              120 numbers that have \"3\" in the \"ones\" position;\r\n" );
document.write( "              120 numbers that have \"4\" in the \"ones\" position;\r\n" );
document.write( "              120 numbers that have \"5\" in the \"ones\" position;\r\n" );
document.write( "              120 numbers that have \"6\" in the \"ones\" position.\r\n" );
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document.write( "        If you do sum all these 720 numbers, you will get the contribution to the sum from the \"ones\" position  \r\n" );
document.write( "        of the value (1+2+3+4+5+6) = 21 repeated 120 times.\r\n" );
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document.write( "    2.  you will have 120 numbers that have \"1\" in the \"tens\" position;\r\n" );
document.write( "              120 numbers that have \"2\" in the \"tens\" position;\r\n" );
document.write( "              120 numbers that have \"3\" in the \"tens\" position;\r\n" );
document.write( "              120 numbers that have \"4\" in the \"tens\" position;\r\n" );
document.write( "              120 numbers that have \"5\" in the \"tens\" position;\r\n" );
document.write( "              120 numbers that have \"6\" in the \"tens\" position.\r\n" );
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document.write( "        If you do sum all these 720 numbers, you will get the contribution to the sum  from the \"tens\" position \r\n" );
document.write( "        of the value (1+2+3+4+5+6)*10 = 21*10 repeated 120 times.\r\n" );
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document.write( "    3.  you will have 120 numbers that have \"1\" in the \"hundreds\" position;\r\n" );
document.write( "              120 numbers that have \"2\" in the \"hundreds\" position;\r\n" );
document.write( "              120 numbers that have \"3\" in the \"hundreds\" position;\r\n" );
document.write( "              120 numbers that have \"4\" in the \"hundreds\" position;\r\n" );
document.write( "              120 numbers that have \"5\" in the \"hundreds\" position;\r\n" );
document.write( "              120 numbers that have \"6\" in the \"hundreds\" position.\r\n" );
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document.write( "        If you do sum all these 720 numbers, you will get the contribution to the sum from the \"hundreds\" position  \r\n" );
document.write( "        of the value (1+2+3+4+5+6)*100 = 21*100 repeated 120 times.\r\n" );
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document.write( "    4.  Similar for \"thousands\" position;\r\n" );
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document.write( "    5.  Similar for the \"ten thousands\" position;\r\n" );
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document.write( "    6.  Similar for the \"hundred thousands\" position.\r\n" );
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document.write( "Finally, you will have in the total sum \r\n" );
document.write( "         the contribution of 21*120          from the \"ones\" position; \r\n" );
document.write( "         the contribution of 21*120*10       from the \"tens\" position; \r\n" );
document.write( "         the contribution of 21*120*100      from the \"hundreds\" position; \r\n" );
document.write( "         the contribution of 21*120*1000     from the \"thousands\" position; \r\n" );
document.write( "         the contribution of 21*120*10000    from the \"tens thousands\" position; \r\n" );
document.write( "         the contribution of 21*120*100000   from the \"hundred thousands\" position. \r\n" );
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document.write( "Now sum 6 lines above, and you will get the final answer, which I wrote at the very beginning.\r\n" );
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