document.write( "Question 1130162: Using the digits 0,1,2,3,4,6 and 7. How many 3-digit even numbers between 200 and 600 can be made? What will be their sum? \n" ); document.write( "
Algebra.Com's Answer #747363 by Edwin McCravy(20055) $\"\"$ $\"About$
You can put this solution on YOUR website!
The solution above is WRONG! \r\n" );
document.write( "Here's the correct solutions, for both inclusive and exclusive of 200 and 600:\r\n" );
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document.write( "Using the digits 0,1,2,3,4,6 and 7, how many 3-digit even numbers between\r\n" );
document.write( "200 and 600 can be made?\r\n" );
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document.write( "You didn't state whether it was inclusive or exclusive of 200 and 600,\r\n" );
document.write( "If it's 'exclusive' the smallest even number allowed is 202, and the largest\r\n" );
document.write( "even number allowed is 576.  If 'inclusive' both 200 and 600 will be included.\r\n" );
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document.write( "However, for calculation purposes, it will be more convenient to include\r\n" );
document.write( "200 and exclude 600. Then we can adjust the count or sum so as to\r\n" );
document.write( "include or exclude 200 and 600.\r\n" );
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document.write( "So we will first assume that the smallest even number allowed is 200 and\r\n" );
document.write( "the largest is 476. That is, we are including 200 but excluding 600.  We\r\n" );
document.write( "will adjust later.\r\n" );
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document.write( "The hundreds digit can only be 2,3 or 4.\r\n" );
document.write( "So that's 3 ways to choose the hundreds digit.\r\n" );
document.write( "The tens digit can be any of the 7 digits 0,1,2,3,4,6 or 7\r\n" );
document.write( "So that's 7 ways to choose the tens digit.\r\n" );
document.write( "The ones digit must be 0,2,4 or 6, to make sure it's even.\r\n" );
document.write( "So that's 4 ways to choose the ones digit.\r\n" );
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document.write( "So the answer we get here is 3•7•4 = 84 ways.\r\n" );
document.write( "Now that includes the number 200, which we are excluding,\r\n" );
document.write( "So we must subtract one from the 84.\r\n" );
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document.write( "Answer: 83, assuming we are not counting 200 or 600.\r\n" );
document.write( "If we are including them both then the answer would be 2 more than 83, or 85.\r\n" );
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document.write( "What is their sum?
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document.write( "Think of the long column of 84 3-digit numbers to be added.  For\r\n" );
document.write( "convenience, let's INclude 200 but EXclude 600.\r\n" );
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document.write( "  200\r\n" );
document.write( "  202\r\n" );
document.write( "  204\r\n" );
document.write( "  206\r\n" );
document.write( "  ...\r\n" );
document.write( "  470\r\n" );
document.write( "  472\r\n" );
document.write( "  474\r\n" );
document.write( " +476\r\n" );
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document.write( "  sum\r\n" );
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document.write( "The 84 hundreds digits in the leftmost column contain an equal number of the 3\r\n" );
document.write( "digits 2, 3 and 4. Since 84/3 = 28. the sum of all the hundreds digits is 28(2+3+4) = 252.\r\n" );
document.write( "Therefore the sum of all hundreds digits contributes 252∙100 or 25200 to\r\n" );
document.write( "the final sum.\r\n" );
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document.write( "The 84 tens digits in the middle column contain an equal number of all 7 digits\r\n" );
document.write( "0,1,2,3,4,6 and 7.  Since 84/7 = 12. the sum of all the hundreds digits is 12(0+1+2+3+4+6+7) = 276.\r\n" );
document.write( "Therefore the sum of all tens digits contributes 276∙10 or to the sum.\r\n" );
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document.write( "The 84 ones digits in the rightmost column contain an equal number of the 4\r\n" );
document.write( "digits 0,2, 4 and 6. Since 84/4 = 21. the sum of all the hundreds digits is\r\n" );
document.write( "21(0+2+4+6) = 252.\r\n" );
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document.write( "Therefore the sum of all ones digits contributes 252 to the final sum.\r\n" );
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document.write( "Therefore the sum which includes 200 but excludes 600 is 25200+2560+252 = 28212.\r\n" );
document.write( "(Chances are this is NOT the answer you want, but it is the easiest to calculate\r\n" );
document.write( "and from which the other two possible answers are easily calculated.]\r\n" );
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document.write( "If we want the sum which excludes both 200 and 600 we must subtract 200 from\r\n" );
document.write( "28212 and get 28012.\r\n" );
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document.write( "If we want the sum which includes both 200 and 600 we must add 600 and get\r\n" );
document.write( "28812.\r\n" );
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document.write( "Edwin \n" );
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