document.write( "Question 262673: How many three-digit numbers are there such that no two adjacent digits of the number are consecutive \n" ); document.write( "
Algebra.Com's Answer #193549 by Edwin McCravy(20056)\"\" \"About 
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document.write( "There are 900 3-digit numbers, for we can choose the first digit\r\n" );
document.write( "9 ways, the 2nd digit 10 ways, and the 3rd digit 10 ways.\r\n" );
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document.write( "So we will subtract from 900 all the 3-digit numbers that\r\n" );
document.write( "have at least one pair of consecutive digits.\r\n" );
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document.write( "Let A = the set of all 3 digit numbers with the first two digits\r\n" );
document.write( "consecutive.\r\n" );
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document.write( "Let B = the set of all 3 digit numbers with the second two digits\r\n" );
document.write( "consecutive.\r\n" );
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document.write( "Then we will use\r\n" );
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document.write( "N(A or B) = N(A) + N(B) - N(A and B) \r\n" );
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document.write( "N(A) =\r\n" );
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document.write( "N(3-digit integers with the first two digits consecutive) \r\n" );
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document.write( "1. There are 8 ways to choose the first digit as 1 through 8,\r\n" );
document.write( "There are then 2 ways to choose the second digit consecutive with the\r\n" );
document.write( "first digit. \r\n" );
document.write( "There are then 10 ways to choose the third digit.\r\n" );
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document.write( "That's 8x2x10 = 160\r\n" );
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document.write( "2. There is 1 way to choose the first digit as 9,\r\n" );
document.write( "There is then 1 way to choose the second digit consecutive with the\r\n" );
document.write( "first digit, namely to choose it as 8 \r\n" );
document.write( "There are then 10 ways to choose the third digit.\r\n" );
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document.write( "That's 1x1x10 = 10\r\n" );
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document.write( "So N(A) = 160+10 = 170 ways,\r\n" );
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document.write( "N(B) =\r\n" );
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document.write( "N(3-digit integers with the last two digits consecutive)\r\n" );
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document.write( "1. There are 8 ways to choose the last digit as 1 through 8,\r\n" );
document.write( "There are then 2 ways to choose the second digit consecutive with the\r\n" );
document.write( "last digit. \r\n" );
document.write( "There are then 9 ways to choose the first digit.\r\n" );
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document.write( "That's 8x2x9 = 144\r\n" );
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document.write( "2. There are 2 ways to choose the last digit as 0 or 9,\r\n" );
document.write( "There is then 1 way to choose the second digit consecutive with the\r\n" );
document.write( "last digit \r\n" );
document.write( "There are then 9 ways to choose the first digit.\r\n" );
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document.write( "That's 2x1x9 = 18\r\n" );
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document.write( "Or a total of 144+18 = 162 ways,\r\n" );
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document.write( "N(A and B)\r\n" );
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document.write( "N(3-digit integers with both the first two and the last two digits\r\n" );
document.write( "consecutive) =\r\n" );
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document.write( "1. There are 6 ways to choose the first digit 2 through 7\r\n" );
document.write( "There are then 2 ways to choose the second digit consecutive with\r\n" );
document.write( "the first digit. There are then 2 ways to choose the third digit\r\n" );
document.write( "consecutive with the second digit.\r\n" );
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document.write( "That's 6x2x2 = 24 ways.\r\n" );
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document.write( "2. There are 3 ways to choose the first digit 1, namely 101, 110, and 121\r\n" );
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document.write( "3. There are 3 ways to choose the first digit 8, namely 878, 879, and 898\r\n" );
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document.write( "4. There are 2 ways to choose the first digit 9, namely 987 and 989.\r\n" );
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document.write( "That's 8 ways.\r\n" );
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document.write( "So N(A and B = 24+8 = 32 \r\n" );
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document.write( "Therefore  N(A or B) = N(A) + N(B) - N(A and B) = 170 + 162 - 32 = 300,\r\n" );
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document.write( "Therefore the answer to the problem is 900 - 300 or 600.\r\n" );
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document.write( "Edwin
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