document.write( "Question 775339: There are 900 three-digit integers. The number of three-digit integers having at least one repeated digit is also a three-digit number. If that number is represented by abc where each letter is a digit compute a-b+c. \n" ); document.write( "

Algebra.Com's Answer #472858 by KMST(5328) $\"\"$ $\"About$

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I may be missing something but here is my calculation.
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\n" ); document.write( "Zeros are a problem, because numbers sequences odf digits like 001 or 011 are not 3-digit numbers. So it is best to treat zero separately.
\n" ); document.write( "If zero is one of the digits, it could appear twice, necessarily at the end, in $\"highlight%28red%289%29%29\"$ 3-digit numbers: 100, 200, ...800, and 900.
\n" ); document.write( "It could also appear once at the end in an additional $\"highlight%28green%289%29%29\"$ 3-digit numbers: 110, 220, ...880, and 990.
\n" ); document.write( "It could also appear once in the middle in an additional $\"highlight%28blue%289%29%29\"$ 3-digit numbers: 101, 202, ...808, and 909.
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\n" ); document.write( "There are also $\"highlight%289%29\"$ 3-digit numbers made of the same repeating digit: 111, 222, ... 888, 999.
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\n" ); document.write( "Besides the $\"red%289%29%2Bgreen%289%29%2Bblue%289%29%2B9=highlight%2836%29\"$ number we already listed, there are 3-digit numbers made of two non-zero digits, digit x appearing twice, and digit y appearing only once.
\n" ); document.write( "There are 9 ways to choose x, and for each of those there are 8 ways to choose y, for a total of $\"9%2A8=72\"$ ordered pairs (x,y).
\n" ); document.write( "For each ordered pair, there are 3 positions where we can place y: at the beginning, in the middle, or at the end. That gives us
\n" ); document.write( " $\"72%2A3=216\"$ 3-digit numbers made of two non-zero digits.
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\n" ); document.write( "If I haven't missed any, and I have not counted any number more than once, that gives me
\n" ); document.write( " $\"216%2B36=highlight%28252%29\"$ three-digit integers having at least one repeated digit.
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\n" ); document.write( "With $\"system%28a=2%2Cb=5%2Cc=2%29\"$ I get $\"a-b%2Bc=2-5%2B2=highlight%28-1%29\"$ \n" ); document.write( "

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