document.write( "Question 1196433: How many numbers between 10 and 1000 have a digit sum of 8?
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Algebra.Com's Answer #829260 by ikleyn(52790) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "The meaning of the word \" between \" in Math problems is not precisely defined.\r
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\n" ); document.write( "\n" ); document.write( "So, I will re-formulate the problem in this form\r
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document.write( "    |   How many numbers from 10 to 999 inclusive have a digit sum of 8?  |\r\n" );
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document.write( "For a minute, I will change my problem and ask\r\n" );
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document.write( "    |   How many numbers from 0 to 999 inclusive have a digit sum of 8?    |\r\n" );
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document.write( "Then it is the same as to ask\r\n" );
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document.write( "   \r\n" );
document.write( "    How many integer solutions does this equation have \r\n" );
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document.write( "         +  +  = 8\r\n" );
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document.write( "    in integer non-negative numbers   >= 0,   >= 0,   >= 0 ?\r\n" );
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document.write( "The answer is known from Combinatorics ( \" Stars and Bars method \" ),\r\n" );
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document.write( "( see this Wikipedia article  \r\n" );
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document.write( "https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)#:~:text=In%20the%20context%20of%20combinatorial,his%20classic%20book%20on%20probability. )\r\n" );
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document.write( "The number of solution in this case is   =  =  = 45.\r\n" );
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document.write( "From it, we should subtract 1, which represents the number \"8\",\r\n" );
document.write( "(we included it, when extended the set from [10,999] to [0,999]).\r\n" );
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document.write( "After subtracting, we get the final \r\n" );
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document.write( "ANSWER.  There are 45-1 = 44  integer numbers from 10 to 999 (inclusive), such that the sum of their digits is 8.\r\n" );
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